Tính chất trao đổi cho môđun và vành - Luận án tiến sĩ Pace Peterson Nielsen

The Exchange Property for Modules and Rings by Pace Peterson Nielsen B. (Brigham Young University, Utah) 2001 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the GRADUATE DIVISION of the UNIVERSITY of CALIFORNIA, BERKELEY Committee in charge: Professor Tsit Yuen Lam, Chair Professor George Bergman Professor Deborah Nolan Spring 2006 UMI Number: 3228440 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted.

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ProQuest Information and Learning Company 300 North Zeeb Road P. Box 1346 Ann Arbor, MI 48106-1346 The Exchange Property for Modules and Rings Copyright 2006 by Pace Peterson Nielsen Abstract The Exchange Property for Modules and Rings by Pace Peterson Nielsen Doctor of Philosophy in Mathematics University of California, Berkeley Professor Tsit Yuen Lam, Chair The exchange property was introduced in 1964 by Crawley and Jonsson. Our work focuses on the problem they posed of when the finite exchange property implies the full exchange property. The methods of this work are two-fold.

The first chapter presents a ring-theoretical construction showing that modules whose endomorphism rings are strongly 7-regular or strongly clean have No-exchange. These methods are further refined to show that modules with Dedekind-finite, regular endomorphism rings have full exchange. The second chapter focuses on direct sum decompositions for modules. We gen- eralize a theorem of Stock, which shows that projective modules with No-exchange have full exchange.

We further show that one may “mod out” by the radical, when considering the exchange property for projective modules. Professor Tsit Yuen Lam. Dissertation Committee Chair To my wife, Sarah, for her patience and love. Contents S1 Notations and Conventlons.

Chapter 1 From a Ring-theoretic Point of View 83 N-Exchange Rings. kg k k vo 10 §4 From 2-Exchange to Finite Exchange .0,4 16 §5 From Finite Exchange to Countable Exchange. 22 $6 From Finite Exchange to Full Exchange ., 29 $7 Lifting through the Jacobson Radical. 36 §8 Examples of Full Exchange Rings.

40 810 Abelian Rings and Commutative Rings. From a Module-theoretic Point of View 47 §11 A Theorem Derived from a Result of Kaplansky. 48 §12 Property (N)g 5D §13 Removing the Radical. 2 en 58 §14 Final Remarks 64 Bibliography 65 ll Acknowledgements I want to thank Tsit Yuen Lam and George Bergman, for carefully reading my work and suggesting many interesting avenues of future research.

I also wish to thank Alex Dugas for reading early manuscripts and for his kind remarks. lil 1 Notations and Conventions By ring we will mean an associative ring with 1. By module we will mean a unital module over a ring, usually with the ring not made explicit and acting on the right. Let M be a module and J an indexing set.

We write MTM) for the direct sum of Ï copies of M, and M! for the direct product of J copies. Endomorphisms of right modules will be written on the left. If a submodule N € M is a direct summand we write N C® M. Some introductory texts on rings and modules include [7] and Suppose we have two modules N C M.

We say N is essential in M if for every non-zero submodule A C M we have NN A # (0). Dually, we say N is small in M if the equation M = K + N implies K = M. In the first case we write N C, M, and in the latter case we write N C, M. A submodule N C M is said to be mazimal if N # M and NC X CM with X #N implies X = M.

The radical of M is defined to be the intersection of all maximal submodules of ă, written as rad(M). If there are no maximal submodules then this is the intersection of the empty family and we put rad(M) = M. It is well known that rad(M) is also equal to the union of all small submodules of M. Let R be aring.

We write J(R) for the Jacobson radical of R. In other words J(R) is the intersection of all the maximal left (or right) ideals of R. In the Introduction and throughout Chapter 1, all modules will be right k- modules, for an arbitrary ring k, unless specified otherwise. We will often write E = End(M;) and we let R be an arbitrary ring (possibly not associated to any module).

In Chapter 2 we depart from this convention and treat all modules as right R-modules. This is due to the fact that Chapter 2 focuses on projective modules, and the ring which acts on such a module plays a central role. 2 Introduction Introduced in 1964 by Crawley and Jonsson [3| for general algebras, the exchange property has since become an important module-theoretic and ring-theoretic tool. The definition is as follows: Definition 2.

Let Ñ > 0 be a cardinal, let k be a ring, and let M;, be a right k-module. We say that M has the X-erchange property if whenever we have right k-module decompositions, A = M @ N = @,., Ai, for some indexing set J with |Z| <N, then there are submodules 4; C A; with A= M @ (@,,, 44). We say that M has finite erchange if M has Ñ-exchange for all finite cardinals. We say M has full exchange if M has Ñ-exchange for all cardinals.

Let k be a division ring and let M be a vector space over k, contained in another vector space A. Suppose further that A has a direct sum decomposition A = @,. It is straightforward to check that M has full ex- change by extending a basis for M to a basis for A, by using only vectors in the submodules A;. In the definition of Ñ-exchange, note that due to the modularity law we have A’ C® A; for eachi € I.

Also note that the exchange property passes to isomorphic modules. If Mf has X-exchange then M has N’-exchange for all 8’ < &. The exchange property behaves well in terms of direct summands and finite direct sums. To prove this we first need the following result: Lemma 2.

If M has |I|-erchange and A= MEN@X = (®,.¡ Ai) @ X then there are submodules A C A; with A= M @ (@,. Let p: A— M @ N be the natural projection from A onto M @ N, with kernel X. Since ker(p) = X we have pl4, : A; —> B; is an isomorphism for each 1 € J, and MON = @,. By the |J|-exchange property, there exist Bi C® B, for each i € I, with M@N = M & (®¿„, Bi).

Now set Al = na, Œị), and so (@jc; 4?) ® X = (Mic; Bi) ® X. Then it is clear that A=M@Ne@X=Múe@ (@z) @X=M@ (@4) SX. tel tel LI Lemma 2. Jf M and M' have the \Ñ-exzchanqe property, then M @ M' has the N-exchange property.

Further, ifP C® M then P also has the X-exchange property. We identify M and M’ with their images in the (outer) direct sum MOM’, and hence consider this as an inner direct sum. Suppose A= (M@M')@N = @,<-, Ai with |J| < &. Then, using the Ñ-exchange property on M, we have submodules 4; C A; with A = M @ (Q@,,, Aj).

But then, using the N-exchange property on M’, and Lemma 2.3 with X = M, we have submodules A? C 4; with A=M@M'@ (ea) 1eL For the second half of the lemma, write M = P®Q, and suppose A= P@N = @,¿¡ Ai with [J] < 8. We may assume, without loss of generality, QM A = (0). Now AeQ=MeN=Qe (@2) ie. and so, using the N-exchange property of /, there are submodules A‘ C A;, and Q © Q, with A@Q=M@N=MeQ@e (@4) =PeQe@@œ (@+) ¡e1 ¡el Consequently we see Q’ = 0.

H Crawley and Jónsson asked the following fundamental question that is still open: Question 2. If a module has 2-exchange, does it have full exchange? It turns out that 2-exchange is equivalent to finite exchange. We will give a straightforward proof in a later section. The question of whether finite exchange implies full exchange, or even No-exchange, can be answered affirmatively for large classes of modules.

One nice way to find such classes is by expressing the X- exchange property in terms of equations in the endomorphism ring of the module. For this we need the following notion: Definition 2. Let M be a module and set # = End(M). Fix a family {z;};er of elements z; € # (not necessarily distinct).

We say that this family is summable if for each m € M the set F,, = {i € I| z;(m) # 0} is finite. Thus, in this situation we have a well-defined element » ri € È with action (x «| (m) = 3` z¡(m). ier i€Fim With this notion in hand, we are ready to prove the following: Proposition 2. Let M be a module.

For any given cardinal, N, the following are equivalent: (1) The module M has the X-exchange property. (2) If we have A=MeEN=QA ¡el with A; M for alli € I, |I| SN, then there are submodules Ai C A; such that A=MeQ@A. iel (3) For every summable family {x;}ier of elements in E = End(M), where the sum is )),-,%; = 1, and \I| SN, there are orthogonal idempotents e; € Ex; with 3,. Since this proposition is used repeatedly, we include the proof for complete- ness.

(2) = (3) : Let J be an indexing set with |I| < 8. Let {z;};er be a summable family in E, summing to 1. Put A = @,„¡ Ai with 4; = M for each i € J. Also, put M' = {a € Ala = (a;(m))ier for some m € M}.

Define x : M — M’ by m+ (xi(m))ier and define y : A — M via (a);er P Ð);cra¿. We see that the map z is an isomorphism, and the inclusion M’ C A splits via # o y. So we have A=MeN=Q4; ¡C1 and hence by (2) there are submodules Ai C® 4;, for each i € J, such that `.) i€l Write A; = Aj @ 47, A’ = @j-, Aj, and A” = @,¿¡ Aj. Let Z be the pro- jection from A to A” with kernel A’, and put y = Yj : MỸ — A”, which is an isomorphism.

Let p; : A” — A! be the natural projections, for each 7 € I. Set e; = yy ‘pix for each i € I. We see that ee) =e ‘pyryy ‘pee = yp piper = 6456. Therefore, these are orthogonal idempotents.

Further, letting z;: A; — 4¿ be the natural projection, with kernel Ai, we have pipx = z¡z¡, and so e; € Ex;. One easily checks that the family {e;};e7 is summable, and Ồ ` iep i = 1, (3) = (1) : Suppose we have A=MeEN=Q4i ier with |JJ < &. Let p: A — M anda; : A — A; be the natural projections corresponding to the respective decomposition. Setting 2; = øm;| for each ¡ € J, we see {Z;};e; is a summable family, and sums to 1.