48IMPLICATIONS OF POSITIVE FORMULAS IN MODULESPHILIPP ROTHMALER

To Miyuki

1. INTRODUCTION: IMPLICATIONSAs is common in logic, given formulas \varphi and \psi , we write \varphiarrow\psi to mean

the sentence \forall\overline{x}(\varphiarrow\psi) , where \overline{x} is a tuple containing all free variables of theformulas \varphi and \psi . We almost always suppress the universal quantifiers when writing
implications.

We adopt the (not unusual) convention that disjunctions V and conjunctions \wedgebind more strongly than arrow.

The most prominent kind of implication may be that of a basic Horn formula,that is an implication \chi of the form \wedge\Phiarrow\psi , where \Phi=\Phi(\overline{x}) is a (possibly infinite)
set of atomic formulas (in the same finitely many variables \overline{x} ) and \psi=\psi(\overline{x}) is asingle atomic formula or\perp (falsum). According to [H], such formulas were shownby McKinsey in 1943 to be preserved in direct products. (It was Horn in 1951,
however, who proved the result in greater generality, whence the name.) Thatis, given a tuple \overline{a} of matching length in a direct product A= \prod_{I}A_{i} , if, for alli\in I , the coordinate tuple \overline{a}(i) of \overline{a} in

A_{i} satisfies \chi , then \overline{a} satisfies \chi in A . The

straightforward proof of this, cf. [ H , Thm.9.1.5], shows more.
Fact 1. If \Phi is a set of formulas preserved in direct factors and \psi is a formula

preserved in direct products, then the implication \wedge\Phiarrow\psi is preserved in direct
products.

Here we say that \Phi is preserved in direct factors if, conversely, the truth of \chi(\overline{a})in A entails that of \chi(\overline{a}(i)) in A_{i} for every i\in I (and every possible direct product).
Recall that a primitive formula is an existentially quantified conjunction of

atomic and negated atomic formulas. Thus a positive primitive (henceforth pp)
formula is an existentially quantified conjunction of atomic formulas. Clearly, pp

formulas are preserved by homomorphisms.

Date: March 26, 2019.

49
PHILIPP ROTHMALER

As is easily seen, pp formulas have both of the features figuring in the hypothesisof Fact 1, cf. [ H , Lemma 9.1.4]. Therefore an implication of the form

\bigwedge_{\varphi\n\Phi}\varphiar ow\psi,with all \varphi and \psi pp (and in the same variables again), is preserved in direct prod‐ucts.

Following [PRZI], we call such implications A ‐formulas—for reasons that become
clear when we look at absolutely pure modules below. (In fact, in that paper only
A ‐sentences were considered, by which we mean A ‐formulas with all free variables

universally quantified out.)
Fact 2. A ‐sentences are preserved in direct products.

Actually, [PRZI] dealt only with modules (over an arbitrary associative ring with1), which is also the context of choice here. As atomic formulas are simply linear
equations in this context, positive primitive formulas are existentially quantified

finite systems of (homogeneous!) linear equations. In a module pp formulas thus
define projections of solution sets of finite systems of linear equations (which always
form additive subgroups). See [P1] for more detail (or [P2], where they are called
pp conditions).
The simplest, and in some sense most prominent, examples of pp formula are
rx=0 and \exists y (x=ry) .

The latter we naturally write as r|x.

Simple examples like nx=0arrow x=0 show that implications of pp formulas neednot be preserved in epimorphic images and are therefore, by Lyndon’s Theorem,

in general not (equivalent to a) positive (formula). But they still behave decently
enough.

On a very general level this can be observed in positive logic as treated by Ben‐

Yaakov and Poizat [BP], where implications of finitary positive (but not necessarily
primitive) formulas play an important role in their development of model theory—for instance, in the proof of the compactness theorem—for the simple reason thatsuch formulas, they call themh ‐inductive,

are preserved in direct limits, as caneasily be verified. This can be used to prove the compactness theorem for setsof h‐inductive sentences in a rather straightforward way [BP, Cor.4]. (Moreover,they prove a preservation theorem saying, conversely, that the theories preserved indirect limits are precisely those that can be axiomatized by h‐inductive sentences[BP, Thm.23].)
If now both antecedent and consequent in an h‐inductive sentence are (positive)
primitive, the implication is preserved in pure epimorphic images (cf. §2.4), whichindicates how close such pp implications get to positive.

50
IMPLICATIONS OF POSITIVE FORMULAS IN MODULES

While also this makes sense for any similarity type of algebraic structures, [Rot2],
it is modules where we know of most fruitful applications of this general model‐

theoretic viewpoint. And so, in this article we survey applications of positive im‐plications (by which we really mean implications of positive formulas) in module
categories. For reasons explained in [PRZI, Thm.4.1], all the implications we deal
with are of the form \wedge\Phiarrow\Psi , where both \Phi and \Psi are allowed to be infinite
sets of pp formulas. Some features turn out nicer, cf. Fact 6 below, if we ‘prepare’\Psi to be closed under finite sums,

so that\Psi becomes\sum\Psi and the implication

turns into

\wedge\Phiarrow\sum\Psi.
Following [PRZI, Thm.4.1], we call such implications symmetric sentences.
We start, in §3, with a list of such implications, some classical, some rather new,and go on, in §4, to exhibit consequences of the corresponding syntactic shapes.
This survey is an extended version of a talk on ‘Incomplete Theories’ given at
the model theory conference at RIMS, December 10‐12, 2018. I wish to thankthe organizer, Professor Hirotaka Kikyo, and RIMS for their hospitality and forproviding the opportunity to present part of this material there.

2. PRELIMINARIES

2.1. Tuples and matrices. Tuples are finite sequences that we think of as column
or row vectors—depending on convenience. For instance, given two tuples 3 and \overline{t}of same length, say k , we use \overline{s}\overline{t} to denote the formal linear combination \sum_{i<k}s_{i}t_{i}.
I.e., we think of matrix multiplication of the appropriate vectors. In other words, wethink of 3 as a row vector (in say S ) and \overline{t} as column vector (in say T , assuming that
there is a product defined between S and T and a sum between those products).
This will always be clear from the context and it will always be assumed that thetuples are of the same length (but will never be mentioned again).
Most of the times, we have that

S is the ring R and T a left

R ‐module.

Sotypically 3 would then be a row vector, while a tuple in a left module is usually
thought of as a column vector.
2.2. Modules and annihilators. Unless indicated otherwise, module means left

module over an associate ring R with 1. The annihilator of X\subseteq R in a module M
is the set ann_{M}X of all elements of M that get annihilated by all elements of X . It
is customary to write \mathfrak{r}(r) instead of ann_{R}RX , also known as the right annihilatorof X in R. (Then t(X) would denote the left annihilator.)

51
51
PHILIPP ROTHMALER

A domain is a ring with no (nontrivial) zero divisors, i.e., a ring in which every
nonzero element has trivial left and right annihilators (i.e., the annihilators are 0
whether considered in RR or in R_{R}. )2.3. Languages and formulas. Some precaution is in order with the term first‐
order, especially when infinitary languages are at play. The original sense of theterm is that the variables can stand only for elements, and hence quantifications

can only be over elements. (Second order languages also have variables for setsof elements, and so on.) So whether infinitary or not, all expressions in L_{\infty\infty} arefirst‐order. I will use the old term elementary to single out the finitary first‐orderpart, i.e., L_{\omega\omega} . (For some reason, over the course of the last half century, model
theorists have replaced ‘elementary’ by ‘first order’—misleadingly so, as Tommy

Kucera has pointed out to me.) The term pp formula is reserved for existentiallyquantified finite conjunctions of atomic formulas, which are elementary.
The pp formulas of a given arity over a ring R—rather their classes modulo
equivalence in all R‐modules—form a lattice with largest element \overline{x}=\overline{x} and smallest

element \overline{x}=\overline{0} , where meet is conjunction and join is ordinary sum (of subgroups).
To see that a conjunction of pp formulas is pp, simply pull out the quantifiers.

For the sum one has to do some more rewriting. First of all, note that, given two
pp formulas of same arity \varphi and \psi , in every module M , the sum \varphi(M)+\psi(M) canbe defined by \exists\overline{yz}(\overline{x}=\overline{y}+\overline{z} \wedge\varphi(\overline{y})\wedge\psi(\overline{z}) . Again, one can pull out all existential
quantifiers to make it look pp.

2.4. Purity. If the ring is a field, hence the modules are vector spaces, all embed‐
dings split, i.e., every subspace is a direct summand. In general, direct summands

may be rare. But the concept of pure submodule plays an intermediate role ingeneral module categories that salvages some of the convenient features of direct
summands while being numerous enough.

For model theorists the easiest definition of purity (and one that applies to any
algebraic structure) is via pp formulas: loosely speaking, an embedding of structures
is pure if it is an elementary embedding with only pp formulas under consideration.

A pure substructure is a substructure whose identical inclusion is a pure embedding.
More precisely, a structure A sitting inside a structure

A\cap\varphi(B)=\varphi(A) for every pp formula \varphi .

B is a pure substructure if(Pp formulas being existential, theinclusion from right to left is, of course, always true.) For modules, it suffices toconsider unary pp formulas, see [P2, Prop.2.1.6]. It is easy to see upon projection,that direct summands are pure submodules (but the converse is far from true: e.g.,
every elementary substructure is pure).

52
IMPLICATIONS OF POSITIVE FORMULAS IN MODULES

A morphism g:Barrow C is said to be a pure epimorphism, [Rotl, §1.6], if for every
pp formula \varphi , every tuple in \varphi(C) has a preimage in \varphi(B) . Again, for modules

1‐place \varphi suffice. (Considering the trivial formula x=x , one sees that such a map
is indeed surjective, thus justifying the term.)
Another convenient feature of modules is that in a short exact sequence

0arrow

Aarrow Barrow Carrow 0 , the monomorphism is pure if and only if the epimorphism is,[Rotl, Cor.3.5]. In this case we speak of a pure‐exact sequence.
Lazard proved that pure‐exact sequences are precisely the direct limits of split

exact sequences, which once again points to their importance (cf. [P2, Prop.2.1.4]
and the references given there).
2.5. Elementary (Prest‐Herzog) duality. Mike Prest [P1] found an an anti‐
isomorphism D , called elementary duality, between the lattice of pp formulas (of agiven arity) for left modules and the lattice of pp formulas (of same arity) for right
modules. Let’s look at it in arity 1. Then D sends r|x to xs=0 and sx=0 to s|x (onthe other side, i.e., to \exists y (x= ys ) ). In particular,

x=x

gets sent to

x=0 and viceversa. Also, meet goes to join and join goes to meet, so that D(\varphi\wedge\psi)=D\varphi+D\psiand D(\varphi+\psi)=D\varphi\wedge D\psi . For the entire definition, see [P2], [Her], or [Rotl].
Ivo Herzog [Her] extended elementary duality to theories (which is why it isalso called Prest‐Herzog duality). This makes it an even more powerful tool. In
particular, the dual of an implication \varphiarrow\psi is defined to be D\psiarrow D\varphi.

Finally, [PRZI] extended all this to certain infinitary formulas. It suffices toknow here that the dual of a symmetric sentence \wedge\Phiarrow\sum\Psi is defined to be\wedge D\Psiarrow\sum D\Phi , which is symmetric again. (Here by the dual of a set I mean theset of the duals.)
An extremely useful application of elementary duality is

Fact 3 (Herzog’s criterion). \overline{a}\otimes\overline{b}=0 in a tensor product A\otimes_{R}B if and only ifthere is a pp formula \varphi (of matching arity) such that \overline{a}\in D\varphi(A) and \overline{b}\in\varphi(B) .
Here we adopt the same convention about tuples: by \overline{a}\otimes\overline{b} we mean the linear
combination of simple tensors, \sum_{i}a_{i}\otimes b_{i}.
3. ALGEBRA

3.1. Classical torsion. Just as for abelian groups, a left module over a domain
R is called

torsionfree if it satisfies all implications

rx=0arrow x=0 ,where r runs

over the nonzero ring elements. So the axioms are a set of implications of atomic

formulas. As the consequent is always the same, one can arrange the entire set into

53
PHILIPP ROTHMALERa single—possibly infinitary—implication of positive formulas, namely,(\vee rx=0)0\neq r\in R^{\cdot}ar ow x=0.
Dually, a left R‐module is called torsion (or periodic in group theory) if it satisfies
the implication

x=xarrow(\vee rx=0)0\neq r\in R^{\cdot}.

In the consequent of this implication one can replace disjunction by sum, for,over a commutative domain, (rx=0+sx=0)arrow(rs)x=0 (which shows that theseaxioms are F ‐sentences in the sense of §4.5).
We observe a curious asymmetry: while torsionfreeness can be axiomatized bya(possibly infinite) set of elementary sentences, namely pp implications, being

torsion is not in general an elementary property—take, for example, the (torsion)
Prüfer group \mathb {Z}_{p}\infty , which is elementarily equivalent to \mathbb{Z}_{p}\infty\oplus \mathbb{Q} , which is mixed.

3.2. Torsion theory. This is done in torsion theory as follows, cf. [S]. One first
defines a map T from R‐Mod, the category of left R‐modules to the category of setsby T(M)= { m\in M : rm

=0 for some0\neq r\in R }, which is easily seen to bea functor. As the so‐called torsion part T(M) forms a subgroup of the underlying
additive group ofM,

it is in fact a functor from

R ‐Mod tothe category Ab of

abelian groups. Now, it is not hard to see that T(M) is even a submodule, making
the map T what is called a preradical.

Given any such preradical, t , one says that M is torsionfree (with respect to t) if
t(M)=0 , and torsion if if t(M)=M . If the preradical t happens to be definable,
these classes are axiomatized by the implications

t(x)arrow x=0,respectively,

x=xarrow t(x).

The preradical T enjoys the extra property that T(M/T(M))=0 , which makes

it a radical. This is the same as to say that the factor module of a module modulo
its torsion part is torsionfree.
The important feature of this radical for us is that it is definable by an infinite

disjunction,

rx=0 ,

so that both extreme properties of torsion and tor‐

sionfree become axiomatized by implications involving that disjunction. We willencounter more such definable radicals.

That the map T is a radical depends heavily on properties of the ring. The proof
that it is is easy when R is a commutative domain. Ore introduced conditions that

54
IMPLICATIONS OF POSITIVE FORMULAS IN MODULES

make sure the same works even when the domain is no longer commutative, cf. [S].
However, there are domains that are not Ore.

Another issue is to extend this to non‐domains. The problem being that thering—as a , say left, module over itself (denoted RR )—is no longer

T ‐torsionfree

if it has zero divisors. Namely, any left zero divisor is a member of T(RR) . One
does not like that. The remedy is to let r run only over so‐called regular elements(which means non‐zero divisors, on either side). If one now puts the Ore conditions
on those one gets the same effect for that adjusted radical.

A torsion theory in the technical sense is then the pair of the two classes of tor‐
sionfree modules and of torsion modules. Every such pair gives rise to a preradical,
whose value on a module M is defined to be the maximal torsion submodule ofM,

see [ S , Ch.VI].
3.3. Hattori torsion. An elegant solution to the aforementioned problems withzero divisors was found by Hattori [Hat]. He defined a module F to be torsionfree,

we will say \mathfrak{h} ‐torsionfree (where \mathfrak{h} stands for Hattori),1 if an element in F can be
annihilated by a ring element r\in R only if it is a linear combination of elements inwhich every coefficient itself is annihilated by r . This can be expressed by infinitary
implications as follows.

Given a tuple 3 in R , let \overline{s}|x denote the pp formula \exists\overline{y}(x=\overline{s}\overline{y}) (remember,with this notation the tuples are assumed to be of matching length).
For the purposes of the next implication, givenr\in R ,write 0(r) for the set\mathfrak{r}(r)^{<\omega} of all (finite) row vectors 3 with entries from \mathfrak{r}(r) . In other words, 0(r) is
the set of all row vectors from R with r\overline{s}=\overline{0}.
Then the above statement about annihilation and linear combinations can be

expressed by the (possibly infinitary) implication

rx=0arrow \vee\overline{s}|x.
\overline{s}\in o(r)Therefore \mathfrak{h} ‐torsionfreeness is axiomatized by all such implications where

r

runs

over all (sic!) of R.
Again, torsion theory (as an algebraic theory) has a standard way of obtaining

a torsion theory (in the technical sense) from a notion of torsionfreeness. Define
T to be\mathfrak{h} ‐torsion if Hom(T, F)=0 for every \mathfrak{h} ‐torsionfree module

the functor from R‐Mod to Ab that singles out, in any

F.

Let \mathfrak{h} be

M\in R ‐Mod, the largest\mathfrak{h} ‐torsion submodule \mathfrak{h}(M) . Hattori [Hat] shows that this is always a preradical andgives conditions when it is a radical, see also [Rot3, §5] for a discussion.
1_{In} the literature, these are often simply called torsionfree, cf. e.g. [P2, §2.3.2].

55
PHILIPP ROTHMALER

Note that \mathfrak{h} ‐torsionfreeness becomes classical torsionfreeness when the ring is adomain (commutative or not).
\overline{s}|x in Hattori torsion de‐

3.4. Flat modules. Note that the disjunction

fines in RR exactly the right annihilator of r . To make things more visible, let \varphi_{r}stand for the pp formula

rx=0 defining it.

As mentioned, in RR , the disjunction
in an arbitrary module

M it defines

M generated by all products

rm\varphi_{r}(_{R}R) M ,\overline{s}|x simply defines \varphi_{r}(_{R}R). Butthat is, the additive subgroup ofwith r\in\varphi_{r}(_{R}R) and m\in M , hence exactly thegroup of all linear combinations in M with coefficients in \varphi_{r}(RR) . In other words, M
satisfies the implication rx=0

So

M is

arrow\overline{s}|x precisely when \varphi_{r}(M)\subseteq\varphi_{r}(RR)M.

\mathfrak{h} ‐torsionfree if and only if it satisfies these implications for all r\in R,

if and only if \varphi_{r}(M)\subseteq\varphi_{r}(RR)M for all r\in R. (Note, the reverse implication isalways true, in any module.)
Requiring this inclusion\varphi(F)\subseteq\varphi(RR)F
for every pp formula \varphi one obtains the notion of flat module F—by a result ofZimmermann [Zim], see [P1] and [P2, Thm.2.3.9]; but for the purposes at hand, we
may adopt this as the definition. (And again, it is easily seen that the inclusionfrom right to left is always true.)
From this one sees at once that every flat module is \mathfrak{h} ‐torsionfree. Over a com‐mutative principal ideal domain, in particular, for abelian groups, the converse istrue. More generally, the same holds for RD‐rings, see [P2, 2.4.16].
To produce an axiomatization of flatness by infinitary implications, first gener‐
alize the previous notation to an arbitrary pp formula \varphi by writing 0(\varphi) for the

set \varphi(_{R}R)^{<\omega} of all (finite) row vectors 3 with entries from \varphi(_{R}R) and consider theinfinite disjunction\overline{s}|x , which indeed defines \varphi(_{R}R) M in every module M.

It is now clear that a module is flat if and only if it satisfies all the implications\varphiarrow \vee \overline{s}|x,\overline{s}\in o(\varphi)

where \varphi runs over all pp formulas (and again, 1‐place \varphi turn out to suffice).
The union the disjunction defines in a module is exactly the sum of the subgroups

defined by the disjuncts (for torsionfreeness, simply concatenate the tuples 3, but
this is as clear for arbitrary \varphi , since, adding two disjuncts, one simply gets longer

linear combinations of the same kind). Thus we finally obtain an axiomatization of

56
IMPLICATIONS OF POSITIVE FORMULAS IN MODULES

flatness by the symmetric sentences\varphiar ow\sum_{\overline{s}\ino(\varphi)}\overline{s}|xfor every (unary) pp formula \varphi.
3.5. Purely generated modules. Here I discuss an axiomatization result from[HR], which generalizes that for flat modules and for which some more backgroundis needed.

It can be easily understood that in the ring R , as a left module over itself, every
elementr

not only satisfies the pp formula r|x , but that this is the smallest pp

formula it satisfies, i.e., for every unary pp formula \varphi with r\in\varphi(RR) , one has

r|x\leq\varphi . We say, r|x generates the pp type of r in RR . Clearly, this passes on to
direct powers of RR , that is, to free modules, except one needs to allow divisibility

formulas \overline{r}|x with tuples as divisors, as before. And finally it passes down to direct
summmands of free modules, which is to say, to projective modules. This can bedone for tuples as well, and one obtains that pp types in projective modules are

generated by divisibility formulas, [P2, Lemma 1.2.29].
More can be said. If \overline{a} is a tuple in a projective module

P whose pp type isgenerated by such a pp formula \varphi , then this tuple actually satisfies this formula

freely: we say (P, \overline{a}) is a free realization of \varphi if \overline{a} satisfies (or realizes) \varphi in P and,
whenever a tuple \overline{b} in a module M satiesfies \varphi as well, then there is a homomorphismf :Parrow M

sending \overline{a} to \overline{b} . It is an easy exercise to show, this implies that \varphi

generates the pp type of \overline{a} in P.
Modules in which every tuple freely realizes some divisibility formula are called
locally projective. If we allow arbitrary pp formulas as generators, we obtain theconcept of locally pure projective or strict Mittag‐Leffier module, see [Rot4] for a
detailed discussion.

We say a classCis purely generated by a class \mathcal{B} if every member of C is apure‐epimorphic image of a direct sum of modules from \mathcal{B} . This concept plays animportant role. For instance, by a theorem of Lenzing, the flat modules are purely

generated by finitely generated projectives, cf. discussion in [HR] before Thm. 2.1.
Suppose \mathcal{B} is a class of locally pure projective modules closed under finite direct

sum, and C is the class purely generated by \mathcal{B} . It was shown in [HR, Thm.2.1] thatthen C is axiomatized by implications of the form

\varphiar ow\sum ppf_{\mathcal{B} \varphi,
where ppf_{\mathcal{B} \varphi is the (usually infinite) set of pp formulas below \varphi (in the latticeorder) that are freely realized in some member of \mathcal{B}.

57
PHILIPP ROTHMALER

In [Rot4], more such axiomatizations can be found. (Note, a proper definitionof ppf_{\mathcal{B} \varphi is missing in [HR], but it was alluded to at the end of [HR, Rem.4.2].)
3.6. Injective torsion. Returning to torsion, consider injective torsion as intro‐

duced in [MaRu, Def. 2.1], whose radical \mathfrak{s} is defined by letting the torsion part\mathfrak{s}(M) of M\in R‐Mod be the kernel of the map \varepsilon\otimes_{R}1_{M} , where \varepsilon denotes an in‐jective envelope R_{R}arrow E of R_{R} . Applying elementary duality (§2.5), especially
Herzog’s criterion Fact 3, one can quite easily show, [MaRo] , that \mathfrak{s} is definable bya(possibly infinite) disjunction of pp formulas. More precisely, \mathfrak{s}(M) is the unionof all pp subgroups \psi(M) for which \psi(_{R}R) =0, [MaRo] . This shows that the\mathfrak{s}‐torsion modules are axiomatized by the implication

x=xarrow \mathfrak{s}(x),while the \mathfrak{s} ‐torsionfree modules are axiomatized by the implication\mathfrak{s}(x)arrow x=0.
Here \mathfrak{s}(x) denotes the aforementioned disjunction of formulas.

While this may

involve an infinite disjunction in the antecedent, as with classical torsion the entire
implication is equivalent to the set of pp implications\psi(x)arrow x=0,
where \psi runs over the pp formulas that vanish in RR_{H} . Thus the class of \mathfrak{s} ‐torsionfree

modules is axiomatized by pp implications (and thus elementary). More will be saidin §4.4.

3.7. Positive primitive (pp) torsion. In direct generalization of the behavior
of injective torsion, one may consider the following generalization (introduced in[Rot3]). Given any class \mathcal{F} of left R‐modules, let \mathfrak{s}_{\mathcal{F} (x) be the disjunction of all pp
formulas that vanish on all members of \mathcal{F} . Denote by \mathfrak{s}_{\mathcal{F} the map from R‐Mod tosets that singles out in M\in R‐Mod the subset defined by the disjunction \mathfrak{s}_{\mathcal{F} (x) .

It is quite easy to see [Rot3, Rem.5.5] that the map \mathfrak{s}_{\mathcal{F} is a preradical—namely,
in every module

M , the union defined by the disjunction in question is in factsum, and, moreover, it forms a submodule ofM.

a

The interesting new fact is that it

always is a radical [MaRo] . And so we have two new classes, the \mathfrak{s}_{\mathcal{F} ‐torsion modules
and the \mathfrak{s}_{\mathcal{F} ‐torsionfree modules, axiomatized, respectively, by the implications

x=xarrow \mathfrak{s}_{\mathcal{F}}(x)and\mathfrak{s}_{\mathcal{F}}(x)arrow x=0.

58
IMPLICATIONS OF POSITIVE FORMULAS IN MODULES

For the same reason as in injective torsion, also the class of \mathfrak{s}_{\mathcal{F} ‐torsionfree mod‐ules is elementary.

Note, \mathfrak{s}=\mathfrak{s}_{R}R=\mathfrak{s}_{\mathcal{F}} for \mathcal{F}=Rb , the class of flat left

R ‐modules(the latter

equality follows from the axiomatization of flat modules above).
3.8. Classical divisibility. Let
sionfree axiom

rx=0arrow x=0

R be a domain.

The elementary dual of the tor‐

is x=xarrow r|x .

Doing this for all nonzero ringelements, we see that the elementary dual of the class of torsionfree left R‐modules
is that of divisible right R‐modules. Similarly we see that the class of divisible left
R ‐modules is axiomatized by the set of implications

x=xarrow r|xwith

0\neq r\in R.

3.9. Hattori divisibility. Taking the (infinitary) elementary dual of Hattori’s tor‐
sionfreeness (or \mathfrak{h} ‐torsionfreeness) of §3.3, one obtains Hattori’s definition of divisi‐
bility: a module over an arbitrary ring R is called divisible if it satisfies the following(in general, infinitary) implication(\bigwedge_{sr=0,s\in R}

sx ± 0 ) arrow r|x.

Written in plain language, this means that a module is divisible iff an elementx

is divisible byr\in R whenever the left annihilator of(which is certainly necessary for

x

r

in

R annihilates alsox

to be divisible by ). It is curious to note thatr

Lam in [ L , Def.3.16] makes precisely this definition without quoting Hattori’s work
and without making the dual definition of torsionfreeness.

3.10. Absolutely pure modules. A module is called absolutely pure if it is purein every module that contains it as a submodule. Since direct summands are pure,

an injective module (i.e., a module that constitutes a direct summand in in every
module that contains it) are absolutely pure. (Over noetherian rings, the converse istrue, so one may just think of injectives.) It was shown in [PRZ2, Prop.1.3] (see also
[P2, Prop.2.3.3]) that M is absolutely pure if and only if every pp subgroup is an an‐
nihilator, more precisely, iff for every pp formula \varphi one has \varphi(M)=ann_{M}D\varphi(R_{R}) .

Note, this makes sense, as D\varphi is a right formula and can thus be applied to thering as a right module.

It follows from the easy direction of Herzog’s criterion Fact 3 that the inclusionfrom left to right is always true. Recall from §2.4 that it suffices to consider 1‐place
pp formulas for purity. The same applies to absolute purity. One concludes that

59
PHILIPP ROTHMALER

the class of absolutely pure modules is axiomatized by the implications( \wedge rx=0) arrow \varphi(x),r\in D\varphi(R_{R})one for every \varphi.

These axioms are dual, under the extended elementary duality of [PRZI], tothe axioms of flatness for right modules, see the introduction to §4 in that paper.

This extends an earlier result of Herzog [Her], showing the same for the case thatthese classes are elementary (which, by a result of Eklof and Sabbagh, is the caseprecisely when R is left coherent).
4. LOGIC

Ever since the beginning of model theory (or universal algebra for that matter)-namely since Birkhoff’s preservation theorem—it has been known that the mere

syntactic shape of axioms may have sweeping consequences on the structure of its
model class. We illustrate this by the examples that we have encountered in theprevious section. Most of this holds true in the larger context of structures in any
algebraic signature.
4.1. Valid pp implications: Lemma Presta. It would be a shame to talk about

implications of pp formulas in modules without mentioning a description of thosethat are valid, i.e., true in all left R‐modules—in which case we write \varphi\leq\psi . This

description was found by Mike Prest [Pl, §8.3] and brought to my attention by
Gena Puninski about three decades ago as ‘Lemma Presta.’ I have always foundthe exact statement difficult to memorize, which is why I wish to present it here—in
a way that is easy to remember.

Lemma Presta, [Pl, Lemma 8.10] or [P2, Cor.1.1.16], describes the pp implica‐
tions \varphiarrow\psi that are valid (in R‐Mod), i.e., for which we have \varphi\leq\psi . It shows thatfor any pp formulas \varphi and \psi , one can find a first order sentence in the languageof rings whose truth in

R is equivalent to\varphi\leq\psi . Here is how. Let \varphi and \psi be

written in matrix form as A|B\overline{x} and C|D\overline{x} , respectively.
Write the matrices

X,

A,

B,

C,

Y,

Z

A, B, C with another three unknown matrices X, Y, Z as
and form the products

XA,

XB,

CY,

CZ .

Assumingthat all the matrices match appropriately, state that the outer two products areequal, XA=CZ , and that D is equal to the sum of the inner two, XB+CY=D.

It is an easy exercise in formalization to write the statement that is intended tomean \exists XYZ(XA=CZ\wedge XB+CY=D) as a first order statement in the theoryof rings. Let us prove that this is the desired sentence.

60
IMPLICATIONS OF POSITIVE FORMULAS IN MODULES

Theorem 1 (Lemma Presta).
A|B\overline{x}\leq C|D\overline{x} if and only if R\models\exists XYZ(XA=CZ\wedge XB+CY=D) .
Proof. If the right hand side holds, we have

A|B\overline{x}\leq XA|XB\overline{x}\sim CZ|XB\overline{x}\leq C|XB\overline{x} , and since C|CY\overline{x} is triviallytrue, also A|B\overline{x}\leq C|(XB+CY)\overline{x} , which is the left hand side, as desired.
For the hard direction, consider a moduleA\overline{a}=B\overline{b} , or(A, -B)(\overline{\frac{a}{b} )=\overline{0}M with generators \overline{a}\overline{b} and relations. We will use the standard feature of a module

given by generators 7 and relations G\overline{z}=\overline{0} (where G is a matching matrix over
the ring) that any other matching matrix H annihilating 7 must be divisible on theright by G , i.e., there must be a matching matrix X withNote that \overline{b} satisfies A|B\overline{x} in

M,

H=XG.

hence, by hypothesis, also C|D\overline{x} .

Pick awitness \overline{c} in M such that Ci=D\overline{b} . Now, being in M allows us to write \overline{1} as Z\overline{a}+Y\overline{b},(\overline{\frac{a}b})=\overline{0}so CZ\overline{a}+CY\overline{b}=D\overline{b} , hence ( CZ , CY—D). The standard feature
mentioned before yields a matrix such that ( , CY—D) =X(A, -B) .
X

CZConsequently, XA=CZ and XB+CY=D , as desired.

\squareThis result is an essential ingredient in the development of elementary duality,

cf. [Pl, Thm.8.21], [P2, Prop.1.3.1], or [Rotl, Prop.1.10]. See [Her2, Thm.39] forLemma Presta viewed as a completeness theorem.

4.2. Symmetric sentences. The following is easily verified using the very defini‐tion of purity (that pp formulas pass up and down in extensions).
Fact 4. Symmetric sentences are preserved in pure substructures.
4.3.

A ‐sentences.

Symmetric sentences in which the the consequent consists of asingle pp formula we called

A ‐sentences and saw, in Fact 2, that A ‐sentences arepreserved in direct products. (We are back to infinitary implications: \Phi is allowed
to be an infinite set of pp formulas.)Remembering the axiomatizations of divisibility (§3.9) and absolute purity (§3.10)

by (possibly infinitary) A ‐sentences, one infers that the classes of divisible modules
and that of absolutely pure modules are closed under direct products. This im‐
plies the same for the class of injectives modules, for ‘injective=pure ‐injective+

absolutely pure’ and direct products of pure‐injective modules are pure‐injective.
The question arises how much more can be said: can one extend this to arbitrary

symmetric sentences? The answer is ‘no’ by a variant of McKinsey’s lemma, see[ H , Cor.9.1.7], whose straightforward proof works just as well for pp formulas.

61
61
PHILIPP ROTHMALER

Fact 5 (McKinsey’s lemma). Let \Phi and \Psi be sets of pp formulas.

If\wedge\Phiarrow\vee\Psi is preserved in direct products, then there is a single \psi\in\Psi suchthat\wedge\Phiarrow\vee\Psi is equivalent to\wedge\Phiarrow\psi.
In particular, symmetric sentences closed under product are A ‐sentences.

4.4. Pp implications:definable subcategories. Consider an

A ‐sentenceasabove. If \Phi is finite (or, equivalently, a singleton2), we can write it as a pp im‐
plication, i.e., an implication of the form \varphiarrow\psi with both \varphi and \psi pp formulas.
Having characterized, in Lemma Presta, the pp implications that are always true,

we go on to describe (classes axiomatized by) arbitrary pp implications.

Such sentences are elementary (finitary first‐order), and therefore classes of mod‐els of pp implications are elementary classes. Crawley‐Boevey called such classes, orrather the corresponding full subcategories of modules, definable subcategories (ofR‐Mod).3 Since trivially pp implications are A ‐sentences, definable subcategories

are elementary classes closed under direct products.

The largest definable subcategory of R‐Mod is

R ‐Mod itself.

This is, because

it even is an equational class (or variety in Birkhoff’s terminology of universal

algebra), i.e., a class defined by sentences of the form \exists\overline{x}\alpha with \alpha=\alpha(\overline{x}) a term
equation (hence an atomic formula). Note, such a sentence is equivalent to theimplication \overline{x}=\overline{x}arrow\alpha , clearly a pp implication.

There is a preservation theorem for definable subcategories: a class of modules
constitutes a definable subcategory if and only if it is closed under direct products,

direct limits and pure substructures, [Rot2] or [P2, Thm.3.4.7]. From this it fol‐lows that definable subcategories are exactly the (full subcategories on) additiveelementary classes, i.e., elementary classes closed under finite direct sum and direct

summand, see [HR, §1. 1] again.
Plain inspection of the axioms shows that among the examples discussed in thealgebra section above, the class of torsionfree modules over a domain and the classesof \mathfrak{s}_{\mathcal{F} ‐torsionfree modules (with \mathfrak{s}_{\mathcal{F} pp torsion) constitute definable subcategories.
In particular, the \mathfrak{s} ‐torsionfree modules (for injective torsion \mathfrak{s} ) do.
4.5.

F ‐sentences:

Pure Serre subcategories.

Following [PRZI, §4.2], an F‐

sentence is a symmetric sentence with antecedent a singleton, i.e., an implicationof the form \varphiarrow\sum\Psi with \varphi pp and \Psi a set of pp formulas. An F‐class is a classof structures axiomatized by F ‐sentences.

2_{Finite} conjunctions of pp formulas are pp.
3_{For} a discussion of the history of (and the names for) that concept, see [HR, §1.1].

62
IMPLICATIONS OF POSITIVE FORMULAS IN MODULES

A class K is said to be a pure Serre subcategory if, in every pure‐exact sequence,
the outer terms belong to

K if and only if the middle term does.

In other words,

pure Serre subcategories are closed under pure submodules, pure epimorphic images
and pure extensions.
F ‐sentences

It is easy to see that
epimorphic images.

are preserved in pure submodules and pure

For instance, given a pure epimorphismg:

Barrow C

such

that B\models\varphiarrow\vee\Psi , if \overline{c}\in\varphi(C) , then, by purity, there’s a preimage \overline{b}\in\varphi(B) ,which then must satisfy also some \psi\in\Psi ; finally, any homomorphism preserves pp
formulas, hence \overline{c}\in\psi(C) , which concludes the proof that C\models\varphiarrow\Psi.
The converse is also true:

Fact 6. [HR, Lemma 2.2]
But to show that

F‐classes form pure Serre subcategories.

F ‐sentences

are preserved in pure extensions, though still

straightforward, one needs to make sure that the big disjunction is in fact a sum,i.e., one needs to have the implication in true F ‐form \varphiarrow\sum\Psi , see the details in

the proof of [HR, Lemma 2.2]. The same applies to the next property of F ‐classes—

here one has to add disjuncts for every coordinate in the (finite!) support of anelement in the direct sum.

Fact 7. [HR, Prop. 2.3]

F‐classes are closed under direct sum.

In arbitrary direct products the proof breaks down as one cannot expect tobe able to add up infinitely many disjuncts from \Psi to get a single one. In fact,
McKinsey’s lemma, Fact 5, applies and shows that, under the assumption that
the F ‐sentence is preserved in direct products, a single disjunct must have worked
throughout to begin with:

Fact 8. If an implication \varphiarrow\vee\Psi is preserved in direct products, then there is asingle \psi\in\Psi such that \varphiarrow\vee\Psi is equivalent to \varphiarrow\psi.
In particular, F‐sentences preserved in direct products are pp implications. Hence
F‐classes closed under direct product are definable subcategories.

(This was extended to pure Serre subcategories in [HR, Prop. 2.8].)
Examples of F ‐classes are all classes of torsion modules for the various kindsof torsion discussed in the algebra section (except for possibly Hattori torsion).
Further, the classes of Hattori torsionfree modules, of flat modules and the ones

from §3.5 are also F ‐classes.
REFERENCES[BP] Ben Yaacov, I. and Poizat, B., Fondements de la Logique positive, Journal of Symbolic
Logic, 72 (2007) 1141‐1162

63
PHILIPP ROTHMALER[Hat] Hattori, A., A foundation of torsion theory for modules over general rings, Nagoya Math.
J. 17 (1960) 147‐158.
[Her] Herzog, I., Elementary duality of modules, Trans. A.M.S. 340(1) (1993) 37‐69.
[Her2] Herzog, I., Linear algebra over a ring, arXiv:0909.0436 [math.KT].
[HR] Herzog, I. and Rothmaler, Ph., When cotorsion modules are pure injective, J. Math. Logic,
9 (1) (2009) 63‐102.
[H] Hodges, W., Model Theory, Encyclopedia of Math. and its Applications, vol. 42, Cambridge
University Press 1993.

[L]

Lam, T.Y., Lectures on Modules and Rings, Graduate Texts in Math., vol. 189, Springer
1999.

[MaRo] Martsinkovsky, A. and Rothmaler, Ph., Positive primitive torsion, in preparation.
[MaRu] Martsinkovsky, A. and Russell, J., Injective stabilization of additive functors. II.
(Co)Torsion and the Auslander‐Gruson‐Jensen functor, preprint 2018.
[P1] Prest, M., Model Theory and Modules, London Math. Soc. Lecture Note Series, vol. 130,
Cambridge University Press 1987.

[Pre] Prest, M., Remarks on elementary duality, APAL 62 (1993) 183‐205.
[P2] Prest, M., Purity, Spectra and Localisation, Encyclopedia of Math. and its Applications, vol.
121, Cambridge University Press 2009.

[PRZI] Prest, M., Rothmaler, Ph., and Ziegler, M., Extensions of elementary duality, J. Pure andAppl. Algebra 93 (1994) 33‐56.
[PRZ2] Prest, M., Rothmaler, Ph., and Ziegler, M., Absolutely Pure and Flat Modules and “In‐discrete” Rings, J. Algebra 174 (1995) 349‐372.
[Rotl] Rothmaler, Ph., Mittag‐Leffier modules and positive atomicity,

Habilitationsschrift,

Christian‐Albrechts‐Universität zu Kiel 1994.

[Rot2] Rothmaler, Ph., Purity in model theory, Proc. of Conf. on Model Theory and Algebra,
Essen/Dresden, 1994/95, M. Droste and R. Göbel eds., Algebra, Logic and Application
Series, vol. 9, Gordon& Breach 1997, 445‐469.

[Rot3] Rothmaler, Ph., Torsion‐free, Divisible, and Mittag‐Lefller Modules, Communications in
Algebra, 43:8 (2015), 3342‐3364.
[Rot4] Rothmaler, Ph., Strict Mittag‐Leffler modules and purely generated classes, submitted.
[S]

Stenström, B., Rings of Quotients, Springer‐Verlag, 1996.

[Zim] Zimmermann, W., Rein injektive direkte Summen von Moduln, Comm. Alg. 5(10) (1977)
1083‐1117.