環論シンポジューム (佐藤)On Projective Modulewith unique Maximal Submodule
The 51th Ring and Representation Theory Symposium
Okayama University of Science
Masahisa SatoAichi University & University of Yamanashi

13:10-13:40
September 21, 2018

13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

1/24環論シンポジューム (佐藤)The English version of this lecture was presented in ICRA2018 asfollows;
On Projective Module with unique Maximal Submodule
18th International Conference
On Representations of Algebras (ICRA 2018)
Faculty of Civil Engineering, Czech Technical University
Prague (Czech Republic)
August 13-17, 2018.
This lecture will be done in Japanese.
So please ask your neighbors in case you do not understand Japanese.

13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

2/24Ware’s Problem on Projective modules

Ware’s Problem
Ware’s Problem
Let R be a ring and P a projective right R-module with uniquemaximal submodule L, then L is the largest maximal submodule of P.
R.Ware Endomorphism rings of projective modules, Trans. Amer.
Math. Soc. 155 (1971), 233-256.

Original Problem:
EndR (PR ) is local ring ?
(i.e.) P is completely indecomposable.
13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

3/24

Ware’s Problem on Projective modules

Purpose

PurposeGiving aﬃrmative answer for this problem.

13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

4/24

Ware’s Problem on Projective modules

Key facts
Key facts
W.K. Nicholson, M.F. Yousif, Quasi-Frobenius Rings, Cambridge
University Press (2002).
F.W. Anderson, K.R. Fuller, Rings and Categories of Modules, GTM
13, Springer-Verlag (1992).
Key facts to solve Ware’s problem:(1) Any projective module has a maximal submodule.
This is equivalent to the following fact.
(2) If PJ(R) = P for a projective module P, then P = 0.

13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

5/24Ware’s Problem on Projective modules

RemarkRemarkIf P is finitely generated projective R-module, then the followingconditions are equivalent.
1234

P = eR for some local idempotent e ∈ R.
EndR (P) is local ring.
(i.e.) P is completely indecomposable.
P has unique maximal submodule.
P has the largest maximal submodule.

13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

6/24

Notations

Primitive rings and ideals

Definition 1
R is right primitive ringif R has a faithful simple right R-module.
A two sided ideal T of R is a primitive right ideal
if R/T is a primitive right ring.
An (right) annihilator of S of MR is denoted by
AnnR (S) = {r ∈ R |Sr = 0}.

13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

7/24

Notations

Jacobson radical and primitive right ideal
1. The Jacobson radical J(R)1
2

The intersection of all maximal right ideals of R
The intersection of all primitive right ideals of R

2. What is a primitive right ideal T
1

2
3

A faithful simple right R/T -module R/J is given by the form
T = AnnR (R/J).
T is maximalon two sided ideals included in the above J.
∩∩
I =IT =I ∈ΓI ∈∆Γ = { maximal right ideals I with T ⊂ I }∆ = { maximal right ideals I with R/J ∼= R/I }13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

8/24

Preliminary Lemmas

Basic facts

We keep the follwing notations in this lecture.
P is a projective right R-module with unique maximal submodule L
J is a maximal right ideal such that P/L ∼= R/J
K = AnnR (R/J)

13:10-13:40 September 21, 2018Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

9/24

Preliminary Lemmas

Basic facts
Lemma 1
1
K is maximal among two sided ideals included in J.
2
PK ⊂ L.
3
PI = P for any maximal right ideal I such that R/I ̸∼= R/J,PI = P for any primitive right ideal I ̸= K .
∩∩4
T =Iγ , then PT =PIγ . (Γ is a set of two sided ideals.)
Iγ ∈Γ
Iγ ∈Γ
∩
5
PI = PJ(R). (Γ is the set of primitive right ideals. )I ∈Γ
6L ⊃ PK = PJ(R).

13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

10 /24

Preliminary Lemmas

ExampleIn the R.Ware’s problem, the assumption ”projective” is necessary.
In fact, we give an example of a ring and a module with uniquemaximal submodule but not largest submodule.

13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

11 /
24

Preliminary Lemmas

ExampleExample 1
K is a field
R is a K -algebra with bases {vx |0 ≤ x ≤ 1}
the multiplication vx · vy = vxy .
1
2

R is a uniserial commutative ring.
The ideals of R:
Ji (0 ≤ i ≤ 1) with K -bases {vx |0 ≤ x < i}Ji with K -bases {vx |0 ≤ x ≤ i}. Closure ideal of Ji .

13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

12 /24

Preliminary Lemmas

ExampleAn R-module
M = (R ⊕ J1 )/K {(v0 , 0) − (0, v0 )}has unique maximal but not largest submodule.
In fact,123An R-module
L = (J1 ⊕ J1 )/K {(v0 , 0) − (0, v0 )}is unique maximal submodule of M.
T = (J 1 ⊕ R)/K {(v0 , 0) − (0, v0 )}2
is a submodule of M not included in L.
There is no maximal submodule of M which include T .

13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

13 /24

Structure Theorem

Decomposition
Proposition 2
Let M be a right R-module with unique maximal submodule L. Then1
M is indecomposable
or2
There are direct summands M1 and M2 such that
M = M1 ⊕ M2 ,M1 has unique maximal submodule,M2 does not have any maximal submodules.
Remark 3
By the former example, both cases in the above proposion happen.
13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

14 /24

Structure Theorem

Decomposition proj.
Take a projective module M = P, then we have the following
proposition.
Corollary 4
Let P be a projective right R-module with unique maximal
submodule L. Then1
P is indecomposable
or2
There are direct summands P1 and P2 such thatP = P1 ⊕ P2 ,P1 has unique maximal submodule,
P2 does not have any maximal submodules.
We show the second case does not happen.
13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

15 /24

Structure Theorem

Generalized Nakayama-Azumaya Lemma
Theorem 5
A nonzero projective module has a maximal submodule.
Remark 6
In the proof of the above theorem, we use Axiom choice.
Also we can show this part by using Zorn’s Lemma.
The above theorem is equivalent to the following property.
Theorem 7
Let P be a projective module. Then PJ(R) = P implies P = 0.

13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

16 /24

Structure Theorem

Generalized Nakayama-Azumaya Lemma

Reviewing my proof.
What did I proved ?
The following theorem seems to be proved.
Theorem 8 (Generalized Nakayama-Azumaya Lemma)
Let M be a direct summand of a direct sum of finitely generatedmodules. Then MJ(R) = M implies M = 0.

13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

17 /24

Structure Theorem

Indecomposablity

Theorem 9
Let R be a ring and P a projective right R-module with uniquemaximal submodule L, then P is indecomposable.

13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

18 /24

Structure Theorem

Example of Hinohara

An example ofinfinitely generated indecomposable projective module P
is introduced inS. Hinohara, Projective modules II, The sixth proceeding of Japan
algebraic symposium (Homological algebra and its applications),
Vol.6 (1964), 24―28.
We review this example and we can show that
PJ = P only for one maximal right ideal J.

13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

19 /24

Structure Theorem

Example of Hinohara
Example 2
R : A commutative ring consisting of continuous real functions
with the domain [0, 1].
Maximal ideals : mx = {f ∈ R| f (x) = 0}, (x ∈ [0, 1])
Px : An ideal consisting of f ∈ R with f (t) = 0
for some neighborhood of x
Then
1
Px is infinitely generated indecomposable projective R-module
by S.Hinohara.
2
It is countably generated by Kaplanski.
3
Px does not have a simple factor module isomorphic to R/mx
4
Px has a simple factor module isomorphic to R/my for any y ̸= x(since Px mx = Px and Px my ̸= Px .)13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

20 /24

The solution of Ware’s problemA reduced generator set

An indecomposabel projective module is countably generated
by I. Kaplansky, Projective modules, Ann. of Math. Vol.68 (1958),372–377.
Definition 2
Let {a1 , a2 , · · · } be a generator set of P.
This set is called a reduced generator set if it satisfies
an+1 ̸∈ a1 R + a2 R + · · · + an R
for any n.

13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

21 /24

The solution of Ware’s problem

A reduced generator setLemma 10
Let P be an indecomposable projective R-module. For any nonzero
element a ∈ P, there is a reduced generator set such that a = a1 .
Lemma 11
Let P be a projective right R-module with a maximal submodule L,then we can take a reduced generator set {a1 , a2 , · · · } of P
such that a1 ̸∈ L and ai ∈ L (i ≥ 2).

13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule22 /24

The solution of Ware’s problem

Ware’s problem is trueTheorem 12
Let R be a ring and P a projective right R-module with uniquemaximal submodule L, then
P is isomorphic to eR for some local idempotent e ∈ R.
Particularly L is the largest maximal submodule of P and L = eJ(R).
This is proved by considering M = a1 R + a3 R + · · · and P/M
for some reduced generating set {a1 , a2 , · · · } such thata1 ̸∈ L, ai ∈ L (i > 1) and M ̸= P.

13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

23 /24References

References[1] F.W. Anderson, K.R. Fuller, Rings and Categories of Modules,
GTM 13, Springer-Verlag (1992).
[2] I.N. Hernstein, Noncommutative Rings, The Carus Mathematics
Monographs 15, The Mathematical Association of America, (1968).
[3] S. Hinohara, Projective modules II, The sixth proceeding ofJapan algebraic symposium (Homological algebra and itsapplications), Vol.6 (1964), 24–28.
[4] I. Kaplansky, Projective modules, Ann. of Math. Vol.68 (1958),
372–377.
[5] I. Kaplansky, Fields and Rings, The university of Chicago Press,
Chixago (1969).
[6] W.K. Nicholson, M.F. Yousif, Quasi-Frobenius Rings, Cambridge
University Press (2002).
[7] R. Ware, Endomorphism rings of projective modules, Trans.
Amer. Math. Soc. 155 (1971), 233–256.
13:10-13:40 September 21, 201824 /Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule24References

Thank you for your attention !

13:10-13:40 September 21, 2018
Masahisa Sato (Aichi & Yamanashi)

Proj. mod. with unique maximal submodule

24 /24