Homomorphisms on infinite direct products of groups, rings and monoids

This is the final preprint version of a paper which appeared in Pacific Journal of Mathematics, 274 (2015) 451--495. The published version is accessible to subscribers at http://dx.doi.org/10.2140/pjm.2015.274.451 . HOMOMORPHISMS ON INFINITE DIRECT PRODUCTS OF GROUPS, RINGS AND MONOIDS GEORGE M. BERGMAN Abstract. We study properties of a group, abelian group, ring, or monoid B which (a) guarantee that Q every homomorphism from an infinite direct product A of objects of the same sort onto B factors i I through the direct product of finitely many ultraproducts of the A i (possibly after composition with the natural map B → B/Z(B) or some variant), and/or (b) guarantee that when a map does so factor (and the index set has reasonable cardinality), the ultrafilters involved must be principal. A number of open questions and topics for further investigation are noted. 1. Introduction Q A direct product i∈I A i of infinitely many nontrivial algebraic structures is in general a “big” object: it has at least continuum cardinality, and if the operations of the A i include a vector-space structure, it has at least continuum dimension. But there are many situations where the set of homomorphisms from such a product to a fixed object B is unexpectedly restricted. The poster child for this phenomenon is the case where the objects are abelian groups, and B is the infinite cyclic group. In that situation, if the index set I is countable (or, Q indeed, of less than an enormous cardinality – some details are recalled in then every homomorphism i∈I A i → B factors through the projection Q of i∈I A i onto the product of finitely many of the A i . An abelian group B which, like the infinite cyclic group, has this property, is called “slender”. Slender groups have been completely characterized [21], and slender modules over general rings have been studied. Recent work by N. Nahlus and the author ([5], [6], [4]) on factorization properties of homomorphisms on infinite direct products of not-necessarily-associative algebras (motivated by the case of Lie algebras) have turned up interesting variants on the above sort of behavior. Q First, it turns out that in that context, a useful way to prove every surjective homomorphism i∈I A i → B factors through finitely many of the A i is by proving (a) that every such homomorphism factors through the product of finitely many ultraproducts of the A i , and also (b) that whenever one has a map that factors in that way, the ultrafilters involved must be principal. In this note, we shall consider each of conditions (a) and (b) on an object B as of separate interest. Secondly, we found in [5], [6], [4] that in many cases, though one cannot say that every surjective homo- morphism from a Q direct product to B will itself factor in one of Q these ways, one can say that for every such homomorphism i∈I A i → B, the induced homomorphism i∈I A i → B/Z(B) so factors, where Z(B) denotes the zero-multiplication ideal, {b ∈ B | b B = B b = {0}} (which for B a Lie algebra is the center of B). In the next section, we shall get similar results for groups, with Z(B) Q the center of the group B. (Note that these statements do not say that every surjective homomorphism i∈I A i → B/Z(B) factors as Q stated; such a factorization is asserted only when the homomorphism A → B/Z(B) can be lifted to i i∈I Q a homomorphism i∈I A i → B.) Maalouf [19] abstracts this property, and strengthens some of the results of the papers cited. 2010 Mathematics Subject Classification. Primary: 03C20, 08B25, 17A01, 20A15, 20K25, 20M15, Secondary: 16B70, 16P60, 20K40, 22B05. Key words and phrases. homomorphism on an infinite direct product of groups, rings, or monoids; ultraproduct; slender, algebraically compact, and cotorsion abelian groups. Archived at http://arXiv.org/abs/1406.1932 . After publication, any updates, errata, related references, etc., found will be recorded at http://math.berkeley.edu/~gbergman/papers/ .