THEORY OF RICKART MODULES

THEORY OF RICKART MODULES Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Gangyong Lee, M.S. Graduate Program in Mathematics The Ohio State University 2010 Dissertation Committee: Professor S. Tariq Rizvi, Advisor Professor Ronald Solomon Professor Alan Loper Assistant Professor Cosmin Roman (Co-Advisor)

c Copyright by Gangyong Lee 2010

Abstract This dissertation is devoted to investigations in the theory of Rickart modules. We introduce various notions related to the Rickart property in a general module theoretic setting. Endomorphism ring of a module plays an important role in our study. Topics of our study include: Rickart modules, dual Rickart modules, endoregular modules and endo-rickart modules. These notions constitute the main chapters of the dissertation. A module M is called Rickart if the right annihilator in M of any single element of S = End R (M) is generated by an idempotent of S. This extends the notion of a Baer module as well as that of a right Rickart ring. M is called a dual Rickart module if the image in M of any single element of S is generated by an idempotent of S. M is called an endoregular module if its endomorphism ring is von Neumann regular. M is called an endo-rickart module if the left annihilator in S of any single element of M is generated by an idempotent in S. We provide several characterizations and investigate properties of each of these concepts. We also study the connections of such modules with their endomorphism rings. It is shown that a (dual) Rickart module whose endomorphism ring has no infinite set of nonzero orthogonal idempotents is a (dual) Baer module. We obtain characterizations of well-known classes of rings R, in terms of Rickart R-modules. While direct summands of (dual) Rickart modules are shown to inherit the property, ii

this is not so for direct sums. We obtain conditions which allow direct sums of (dual) Rickart modules to be (dual) Rickart. It is shown that an indecomposable endoregular module has precisely a division ring as its endomorphism ring. We introduce the notion of endo-nonsingularity of a module using the module s endomorphism ring. This notion helps us to obtain new module theoretic versions of the Chatters-Khuri s result. Our investigations allow us to work in a general setting of module theory where several known results on (right) Rickart rings and Baer modules are generalized - sometimes more effectively. iii

This work is dedicated to my mother, Chun Wha Oh. Without her continuous help and support, this work would never have been possible. iv

Acknowledgments I thank my advisor, Professor S. Tariq Rizvi, and co-advisor, Cosmin Roman, both of whom have been very helpful and have shared a lot of time with me during my work. Without them, I can not be one of mathematicians in the world. Professor Rizvi gave me the huge support, patience and unabashed trust and has treated me like his son. Especially, I thank him for suggesting to me this very rich topic that constitutes my thesis and for his guidance throughout this work. Also, I would like to thank Dr. Roman for his help, suggestions, and explaining basic knowledge in a friendly manner. He is a friend and a colleague. I want to thank Professor Jae Keol Park for providing me additional insight, many useful discussions and ideas for further study of mathematics. I want to thank Professor Gary F. Birkenmeier for a number of helpful discussions. My interactions with Professors Park and Birkenmeier have been greatly helpful to me. My thanks go to my sister Kang Eun Lee. Many thanks to professors in the department from whom I took many courses as a graduate student. My thanks also go to my colleagues and members of staff (in particular, Cindy Bernlohr, Denise Witcher) in the Mathematics Department at the Ohio State University, and to the many other people who, in one way or another, helped me through the good and the bad moments during my graduate studies. v

Vita November 21, 1972......................... Born - Seoul, Korea 1996........................................B.S. Sung Kyun Kwan University 1998........................................M.S. Sung Kyun Kwan University 2006........................................M.S. The Ohio State University 2003-present................................ Graduate Teaching Associate, The Ohio State University Research Publications Publications Rickart modules Communications in Algebra, to appear (with S.T. Rizvi and C. Roman) Dual-Rickart modules Communications in Algebra, to appear (with S.T. Rizvi and C. Roman) Direct sums of Rickart modules manuscript submitted (with S.T. Rizvi and C. Roman) Fields of Study Major Field: Mathematics Area of Study: Algebra Studies in Rings and Modules Theory: Advisor: Professor S. Tariq Rizvi; Co-Advisor: Assistant Professor Cosmin Roman vi

Contents Page Abstract....................................... Dedication...................................... Acknowledgments.................................. Vita......................................... ii iv v vi Chapters: 1. Introduction.................................. 1 1.1 Background and Motivation...................... 1 1.2 Summary................................ 7 1.3 Preliminaries.............................. 11 2. Rickart Modules............................... 20 2.1 Definition and Properties of Rickart Modules............ 20 2.2 The Endomorphism Ring of a Rickart Module............ 33 2.3 Rickart Modules versus Baer Modules................ 42 2.4 Direct Sums of Rickart Modules.................... 51 2.5 Free Rickart Modules.......................... 65 2.6 Rickart Modules over a Commutative Dedekind Domain...... 81 3. Dual Rickart Modules............................ 85 3.1 Dual Notion to the Rickart Property of Modules........... 85 3.2 Connections between a Dual Rickart Module and its Endomorphism Ring................................... 96 3.3 Dual Rickart Modules versus Dual Baer Modules.......... 100 vii

3.4 Direct Sums of Dual Rickart Modules................. 107 4. Endoregular Modules............................. 114 4.1 A Generalization of (von Neumann) Regular Rings......... 114 4.2 Indecomposable Endoregular Modules................ 122 4.3 Direct Sums of Endoregular Modules................. 127 5. Endo-Rickart Modules............................ 131 5.1 Definition and Properties of Endo-Rickart Modules......... 131 5.2 The Endomorphism Ring of an Endo-Rickart Module........ 135 5.3 Endo-nonsingular Modules....................... 139 Appendices: A. Characterizations of Some Classes of Rings................. 148 Bibliography.................................... 153 viii

Chapter 1 INTRODUCTION 1.1 Background and Motivation In any ring R, the idempotent elements e 2 = e R play an important role in determining the structure of R because R can be decomposed into smaller pieces in terms of its idempotents, i.e., R = er (1 e)r. Thus, an ability to control the idempotents in some fashion vastly improves our chances of obtaining more information about rings. The notions of Rickart and Baer rings evolved from the connections of idempotents to annihilators in a ring. The usefulness of such an approach is visible in a situation such as attempting to solve a linear equation in one unknown, ax = b with a 0 (or a finite system of such equations). Within a general ring R, it is difficult to gauge existence and especially the uniqueness of a solution to such an equation. If R is a field then the solution is given by x = ba 1 and is unique (and for a system of such equations we can have either no solutions or a unique solution). A domain (a ring where the product of two elements is zero if at least one of the elements is zero) is the next best situation where one can have a practical approach to solving the equation and also have uniqueness of its solution if it exists. The general class of rings in which one could determine all solutions to the original equation (akin to the 1

two cases just presented) includes the class of Rickart rings or Baer rings. We can then ask the natural question: Can we solve a finite system of equations in several unknowns as well? For R a field, the theory of solving systems of linear equations is well established, but for general rings it is not so. Again, a good setting for having some sort of handle on finding all solutions would be to consider rings R such that the direct sums of copies of R satisfy Rickart or Baer property. The notions of Rickart and Baer rings have their roots in functional analysis, with close links to C -algebras and von Neumann algebras. In 1951, Kaplansky defined AW -algebras (C -algebras in which the right annihilator of any subset is generated by a projection). Alternatively put, AW -algebras are C -algebras with the Baer property. Also, in 1955, Kaplansky defined Baer rings and Baer *-rings. A Baer ring (respectively, Baer *-ring) is a ring (respectively, *-ring) in which the right annihilator of any nonempty subset is generated by an idempotent (respectively, a projection). A number of very interesting properties of Baer rings were shown by Kaplansky and further investigated by several other mathematicians. Examples of Baer rings include right self-injective von Neumann regular rings, von Neumann algebras, any domain with a unit element and the endomorphism rings of semisimple modules (thus, the endomorphism rings of all vector spaces). The concept of Baer rings was generalized to that of quasi-baer rings by W.E. Clark [16] in 1967 by replacing the subset by a two-sided ideal in the above definition. Examples of quasi-baer rings include all prime rings, and rings of matrices over Baer rings. It is easy to see that the Baer and the quasi-baer properties are left-right symmetric for any ring. Large classes of rings satisfy the Baer and the quasi-baer properties, respectively. The theory of Baer and quasi-baer rings has come to play an important role, and major contributions to 2

this theory have been made in recent years. Some of the contributors include S.K. Berberian, G.F. Birkenmeier, A.W. Chatters, S.M. Khuri, J.Y. Kim, Y. Hirano, J.K. Park, A. Pollingher, K.G. Wolfson and A. Zaks, among others (see, for example, [4], [6], [7], [8], [9], [10], [11], [15], [38], [53]). In 1946, C.E. Rickart studied C -algebras (i.e, Banach algebras with an involution * such that xx = x 2 ) which satisfy the condition that the right annihilator of every single element is generated by a projection (e 2 = e, e = e). These algebras later came to be named Rickart C -algebras by Kaplansky. In 1960, S. Maeda [36] defined Rickart rings in an arbitrary setting. A ring is called right Rickart if the right annihilator of any single element is generated by an idempotent. A left Rickart ring is defined similarly. In an interesting insight, Hattori [23] introduced in 1960 right p.p. rings, rings with the property that every principal right ideal of R is projective; it was later shown that right p.p. rings are precisely right Rickart rings. Our study of the module theoretic analogue of Rickart rings is based on the original definition of Maeda. The theory of Rickart rings, as a generalization of Baer rings (note that every Baer ring is right and left Rickart), has attracted attention of a number of researchers and major contributions to this theory have been made. The extensive study in the ring theoretical setting until now includes a large number of interesting results by mathematicians such as E.P. Armendariz, S.K. Berberian, G.M. Bergman, S. Endo, M.W. Evans, S. Jøndrup, I. Kaplansky, A.W. Chatters and W.K. Nicholson among others (see, for example, [2], [4], [5], [18], [19], [25], [26], [44]). Examples of right Rickart rings include domains, von Neumann regular rings and 3

right (semi)hereditary rings. In particular, the endomorphism ring of an arbitrary direct sum of copies of a right hereditary ring is a right Rickart ring. In 2004, the notion of Baer rings was placed in the general module theoretic setting by Rizvi and Roman utilizing the endomorphism ring of the module for the first time [40]. It was shown that many results for Baer rings can be proved in the general setting of modules including a type theoretic decomposition similar to the one provided for Baer rings by Kaplansky in [26]. Considering an R-module M as an S-R bimodule where S = End R (M), a module M is said to be Baer if the right annihilator in M of any nonempty subset of S is generated by an idempotent of S (see also [40], [41], [42], [43]). Some examples of Baer modules include Baer rings R viewed as right R- modules, semisimple modules, nonsingular (K-nonsingular) extending modules, free modules of countable rank over a PID. In this body of work, we extend not only the notion of Rickart rings to modules but also generalize the notion of Baer modules and introduce a much larger class of modules. Other than a natural desire to extend the notion of a right Rickart ring to a general module theoretic setting, one of the motivations to study this concept in a module theoretic setting is the question: If R is a right Rickart ring and e 2 = e R, what kind of Rickart property will the right R-module er have? In 1967, L.W. Small [44] proved that a right Rickart ring with no infinite set of nonzero orthogonal idempotents is a Baer ring (Theorem 2.3.2). Consequently, another natural question is: Is there a module theoretic analogue of Small s result? This question is important especially since the notion of Baer modules already exists [40]. Once we have introduced a module theoretic analogue, we will obviously be interested in the question: 4

Do the direct summands and direct sums inherit the property? If not, when will the said property be inherited? It is well-known that the (quasi-)baer property for rings is left-right symmetric. However, a right Rickart ring is not always left Rickart. An example was given by S.U. Chase [14] to distinguish these two classes. We are therefore interested in defining a left-sided notion also. In 1960, S. Endo [18] showed that an abelian ring R is a right Rickart ring if and only if R is a left Rickart ring (Proposition 1.3.26). (A ring R is called abelian if every idempotent of R is central.) Another question that can be asked here is: Can we provide a similar or analogous characterization for abelian modules instead of rings? We will need a module theoretic analogue for a left Rickart ring to do this. The notion of extending modules has been an exciting topic of study in the past decade. As extending modules heavily depend on direct summands, one can expect that such modules will be intrinsically connected to modules having Baer or Rickartlike properties. Recall that a module is extending if every submodule is essential in a direct summand. A ring is called extending if R R is an extending module. This simple property is satisfied by every (quasi-)injective (even uniform) module. Since 1990s, the development of the extending module theory has been a major area of research interest. Even with numerous papers published in the last two decades related to this concept, a number of open problems remain. We hope that our study will provide (partial) answers to some of those questions, given the close connections of our notions to extending modules. In 1980, Chatters and Khuri [15] showed that there are close connections between the Baer rings and the right extending rings. More precisely, they proved 5

that every right nonsingular right extending ring can be characterized as a Baer ring which is right cononsingular (Theorem 1.3.28). In 2004, Rizvi and Roman [40] introduced the notion of K-nonsingularity and proved a module theoretic version of the Chatters-Khuri s result stating that a K-nonsingular extending module is precisely a K-cononsingular Baer module (Theorem 1.3.29). In our study we have continued the research done in Baer module theory in our setting and we were able to extend some of these results related to Baer modules. We also obtain a Chatters-Khuri type of theorem in our work on endo-nonsingular (a module S M R is called endo-nonsingular if {m l S (m) ess SS} = 0 where S = End R (M)). Let R be a ring and define ϕ a : R R, the left multiplication by a map. Then it is well-known that R is a von Neumann regular ring iff ϕ a (R) = ar R R, a R. This was placed in a more general setting by K.M. Rangaswamy [39] in 1967 (see also R. Ware s work [49] in 1971): The endomorphism ring of a module is a von Neumann regular ring iff every kernel and image of endomorphisms are direct summands of the module (Theorem 1.3.27). Also, since von Neumann regular rings play an important role in functional analysis, studying modules which have the von Neumann regular ring as its endomorphism ring is of interest for cross-field researchers, since algebraic methods can sometimes provide alternate ways of proving or understanding operator theory problems. In this work related to Rickart modules, we introduce and study the notions of Rickart, dual Rickart, endoregular and endo-rickart properties for arbitrary modules. Actually, the notion of Rickart modules was introduced by Rizvi and Roman [41] and has also been studied recently (see [33], [34], [35]). We do this by exploiting the connections between a module M and its endomorphism ring S = End R (M). Our 6

investigations answer some of the preceding questions and also provide tools and techniques which develop this theory in the general module theoretic setting where results can be generalized or extended - sometimes with more efficient proofs. 1.2 Summary We begin Chapter 1 with motivation and background. After providing a summary of the dissertation we include preliminary definitions and known results to be used later. In Chapter 2, we introduce the notion of a Rickart module and obtain its basic properties. A module M is called a Rickart module if the right annihilator in M of any single element of S = End R (M) is generated by an idempotent of S, equivalently, the kernel of every endomorphism of M is a direct summand of M. It is shown that every direct summand of a Rickart module inherits the property, that every Rickart module is K-nonsingular, and that it satisfies D 2 condition and the Summand Intersection Property. The endomorphism rings of Rickart modules are investigated. We provide a complete characterization of when is a module with a right Rickart endomorphism ring, also Rickart. Moreover, we also characterize Rickart modules whose endomorphism rings are von Neumann regular. Since every Baer module is a Rickart module we investigate conditions which allow for the converse of this to hold true. A well-known result of L. Small states that a right Rickart ring with no infinite set of nonzero orthogonal idempotents is always a Baer ring. We extend this result to the general module theoretic setting and show that if the endomorphism ring of a Rickart module M has no infinite set of nonzero orthogonal 7

idempotents then M is a Baer module. Furthermore, we investigate indecomposable Rickart modules. Whether or not this module theoretic notion is inherited by direct summands and direct sums is one of natural question. This, even more so since we are motivated by the question about the properties of the right R-module er where e 2 = e in a right Rickart ring R. We show that every direct summand of a Rickart module is Rickart, thus showing that each er is a Rickart R-module if R is right Rickart. However, since the direct sums do not inherit this property, we explore conditions needed for a direct sum of Rickart modules to be Rickart. We first obtain a result on the relative Rickart property and show that if there exists an ordering I = {1, 2,, n} for a class of R-modules {M i } i I such that M i is M j -injective for all i < j I, then n i=1 M i is a Rickart module iff M i is M j -Rickart for all i, j I. As an application, if M is a nonsingular finitely Σ-extending module then M and E(M) are finitely Σ-Baer modules, and E(M) (m) M (n) is a (Baer, hence) Rickart module for any m, n N. Further, we prove that if M i is M j -C 2 for all 1 i, j n, then n i=1 M i is a Rickart module iff M i is M j -Rickart for all 1 i, j n. We obtain characterizations of well-known classes of rings, in terms of Rickart modules. More precisely, we are able to characterize right n-hereditary rings, right semihereditary rings, von Neumann regular rings, right hereditary rings, right V - rings, and semisimple artinian rings. We focus then our investigations on the study of Rickart modules over a commutative Dedekind domain. Indeed, a finitely generated abelian group is Rickart iff it is either semisimple or torsion-free iff it is Baer. 8

The notion of a dual Rickart module is introduced in Chapter 3. We investigate and obtain some of its basic properties and characterizations. A module M is called a dual Rickart (or a d-rickart) module if the image in M of any single element of S = End R (M) is generated by an idempotent of S, equivalently, ϕ S, Imϕ M [32]. We note that when M = R R, the notion of a dual Rickart module coincides with that of a von Neumann regular ring. It is shown that every direct summand of a d-rickart module inherits the property, and that every d-rickart module satisfies C 2 condition and the SSP, and is T -noncosingular. Properties of the endomorphism ring of a d-rickart module are also investigated. We completely characterize a d-rickart module with respect to its endomorphism ring and this ring s inherited property. Also, examples are provided to show that the concept of a d-rickart module is distinct from that of a (Zelmanowitz) regular module. It is shown that every injective right R-module over a right hereditary ring R is a dual Rickart module. Consequently, every injective right R-module over a right hereditary right noetherian ring R is a dual Baer module. Every dual Baer module is a d-rickart module, however the converse is not true, in general. Conditions which allow for this converse to hold true are investigated. Further, it is shown that M is an indecomposable d-rickart module satisfying D 2 condition iff End R (M) is a division ring. We focus on when are direct sums of two or more d-rickart modules, also d- Rickart. We first obtain a result on the relative d-rickart property we introduce, and show that if there exists an ordering I = {1, 2,, n} for a class of R-modules {M i } i I such that M i is M j -projective for all i > j I, then n i=1 M i is a d-rickart 9

module iff M i is M j -d-rickart for all i, j I. Further, we prove that if M i is M j -D 2 for all 1 i, j n, then n i=1 M i is a d-rickart module iff M i is M j -d-rickart for all 1 i, j n. The study of modules whose endomorphism rings are von Neumann regular is our focus in Chapter 4. We call such modules, endoregular modules. Thus R R is an endoregular module iff R is a von Neumann regular ring. It can be checked that every endoregular module is a Rickart as well as a d-rickart module. Thereby, every endoregular module is a Rickart module and a d-rickart module. We also investigate an indecomposable endoregular module which has precisely a division ring as its endomorphism ring. A characterization of an abelian endoregular module as a direct sum of the kernel and image of any endomorphism is provided. We verify that the notion of endoregular modules satisfies natural algebraic properties such as direct summands and direct sums of copies of an endoregular module inherit the property. After introducing the notion of the relative endoregular property, we use it to study direct sums of two or more endoregular modules. Thereby, we are able to prove that a finite direct sum of endoregular modules, which are relatively endoregular to each other, inherits the property. In Chapter 5, the notion of an endo-rickart module is introduced. A module M is called an endo-rickart module if the left annihilator in S = End R (M) of any single element of M is generated by an idempotent in S, equivalently, m M, l S (m) = Se for some e 2 = e S. In addition, R R is an (right) endo-rickart module iff R is a left Rickart ring. It is shown that every fully invariant direct summand of an endo-rickart module inherits the property. Examples are provided to illustrate the concept of an endo-rickart module is distinct from that of a (d-)rickart module. 10

We provide a complete characterization of (Zelmanowitz) regular modules in terms of endo-rickart modules. It is proved that if M is a finitely generated abelian endo- Rickart module then End R (M) is a left Rickart ring. Also, it is shown that a module M is endo-rickart and S S = End R (M) has the SSIP if and only if M is a Baer module. Connections are exhibited between endo-rickart modules and principally extending modules. In particular, we show that every K-nonsingular principally extending module is an endo-rickart module, and every K-cononsingular endo-rickart module is a principally extending module. We introduce the notion of endo-nonsingularity and obtain its basic properties. Even though a direct summand of an endo-nonsingular module is not endo-nonsingular, we show that every fully invariant direct summand of an endo-nonsingular module is endo-nonsingular. It is shown that the endomorphism ring of an endo-nonsingular module is left nonsingular, while the converse is not true. Finally, as an analogue of Chatters-Khuri s result we prove that a module M is endo-nonsingular and End R (M) is a left extending ring iff M is a Baer module and End R (M) is left cononsingular. 1.3 Preliminaries Throughout this dissertation, R is a ring with unity and M is a unital right R- module. For a right R-module M, S = End R (M) will denote the endomorphism ring of M; thus M can be viewed as a left S- right R-bimodule. For ϕ S, Kerϕ and Imϕ stand for the kernel and the image of ϕ, respectively. The notations N M, N M, N ess M, N c M, N M, N M or N M mean that N is a subset, a submodule, an essential submodule, an essentially closed submodule, a fully 11

invariant submodule, a direct summand of M or a fully invariant direct summand of M, respectively. E(M) denotes the injective hull of M and M denotes the quasiinjective hull of M. By R, Q, Z and N we denote the ring of real, rational, integer and natural numbers, respectively. Z n denotes Z/nZ, Z p denotes the p-prüfer group, M (n) denotes the direct sum of n copies of M and Mat n (R) denotes an n n matrix ring over R. We also denote r M (I) = {m M Im = 0}, r S (I) = {ϕ S Iϕ = 0} for I S; r R (N) = {r R Nr = 0}, l S (N) = {ϕ S ϕn = 0} for N M. Definition 1.3.1. A ring R is called a von Neumann regular ring if for any a R, there exists b R such that a = aba. Definition 1.3.2. A Dedekind domain is an integral domain in which every proper ideal is the product of a finite number of prime ideals. Definition 1.3.3. A ring R is called right Rickart if the right annihilator of any single element of R is generated by an idempotent as a right ideal. (This is equivalent to every principal right ideal of R to be projective, i.e., R is a right p.p. ring.) A left Rickart ring is defined similarly. Definition 1.3.4. A ring R is called a Baer ring if the right annihilator in R of any nonempty subset of R is generated, as a right ideal, by an idempotent element of R (in other words, for all I R R, r R (I) = er where e 2 = e R). Remark 1.3.5. The Baer property for rings is left-right symmetric: a ring R is a Baer ring if and only if the left annihilator in R of any right ideal is generated, as a left ideal, by an idempotent element of R. 12

Definition 1.3.6. Let M be a right R-module and let S = End R (M). Then M is called a Baer module if the right annihilator in M of any nonempty subset of S is generated by an idempotent in S. Equivalently, I S, r M (I) = em for some e 2 = e S. Definition 1.3.7. Let M be a right R-module and let S = End R (M). Then M is called a dual Baer module if for every N M, there exists an idempotent e in S such that D(N) = es where D(N) = {ϕ S Imϕ N}. Definition 1.3.8. (C 1 ) Every submodule of M is essential in a direct summand of M. (C 2 ) If a submodule L of M is isomorphic to a direct summand of M, then L is a direct summand of M. (C 3 ) If M 1 and M 2 are direct summands of M such that M 1 M 2 = 0, then M 1 M 2 is a direct summand of M. A module M is called extending if it has (C 1 ), M is called continuous if it has (C 1 ) and (C 2 ), and M is called quasi-continuous if it has (C 1 ) and (C 3 ). Definition 1.3.9. (D 1 ) For every submodule L of M, there is a decomposition M = M 1 M 2 such that M 1 L and L M 2 is small in M. (D 2 ) If L is a submodule of M such that M/L is isomorphic to a direct summand of M, then L is a direct summand of M. 13

(D 3 ) If M 1 and M 2 are direct summands of M such that M 1 +M 2 = M, then M 1 M 2 is a direct summand of M. A module M is called lifting if it has (D 1 ), M is called discrete if it has (D 1 ) and (D 2 ), and M is called quasi-discrete if it has (D 1 ) and (D 3 ). Recall that a submodule N of a module M is said to be small in M if N + L M for any proper submodule L of M. Definition 1.3.10. A module M is called a (Zelmanowitz) regular module if given any m M there exists f Hom R (M, R) such that mf(m) = m. Definition 1.3.11. An idempotent e of a ring R is said to be left (right) semicentral if ae = eae (ea = eae) for all a R. If an idempotent e of a ring R is left and right semicentral then e is called central. A module M is said to be abelian if every idempotent in the endomorphism ring of M is central. A ring R is called abelian if R R is abelian. Definition 1.3.12. A module M R is called nonsingular if the singular submodule of M, Z(M) = {m M mi = 0 where I ess R R }, is zero. A ring R is right nonsingular if R R is nonsingular. A ring R is called right cononsingular if any right ideal, with zero left annihilator, is essential in R R. Definition 1.3.13. A module M is called polyform if for any submodule L M and for any 0 ϕ : L M, Kerϕ is not essential in L, equivalently, if for any submodule L M and for any ϕ : L M, Kerϕ ess L implies ϕ = 0. Definition 1.3.14. A module M is called K-nonsingular if, for all ϕ End R (M), r M (ϕ) = Kerϕ ess M implies ϕ = 0. 14

Definition 1.3.15. A module M is called K-cononsingular if, for all N M, l S (N) = 0 implies N ess M. Equivalently, ϕ(n) 0 for all 0 ϕ S implies N ess M. Definition 1.3.16. A module M is called T -noncosingular if, for every nonzero endomorphism ϕ of M, Imϕ is not small in M. Definition 1.3.17. A module M is called retractable if Hom(M, N) 0, 0 N M ( if N 0 then 0 ϕ S = End R (M) with ϕm N). Definition 1.3.18. A module M is called quasi-retractable if Hom(M, r M (I)) 0, 0 r M (I) for I S S ( if r M (I) 0 then r S (I) 0, I S S). Definition 1.3.19. A module M is said to have the summand intersection property (SIP) if the intersection of any two direct summands is a direct summand of M. M is said to have the strong summand intersection property (SSIP) if the intersection of any family of direct summands is a direct summand of M. M is said to have the summand sum property (SSP) if the sum of any two direct summands is a direct summand of M. M is said to have the strong summand sum property (SSSP) if the sum of any family of direct summands is a direct summand of M. Definition 1.3.20. Let M and N be right R-modules. We say that M is N-injective if, N N and ϕ : N M, ϕ : N M such that ϕ N = ϕ. A module Q is called injective if Q is N-injective for all every R-module N. Lemma 1.3.21. If M i is N-injective, for i = 1,..., n (n N), then n i=1 M i is N-injective. If N is M j -injective, for j = 1,..., n, then N is n j=1 M j-injective. Lemma 1.3.22. (Lemma 7.5, [17]) Let M and N be right R-modules such that M is N-injective. If P M N such that P M = 0, then there exists P P such that M N = M P. 15

Proof. Let π M and π N be the canonical projection of M N onto M and N, respectively. Since P M = 0, π N P is a monomorphism. Define a homomorphism ϕ : π N (P ) π M (P ) by ϕ(π N (x)) = π M (x) for x P. It is easy to check a well-defined homomorphism. Since M is N-injective, there exists a homomorphism ϕ : N M such that ϕ P = ϕ π N. We can construct the following submodule of M N: P = {ϕ(n) + n n N}. It is easy to check that P P. Since P M = 0 because ϕ(n)+n M n M N n = 0, N M +P M N = M +P = M P. Definition 1.3.23. Let M and N be right R-modules. We say that M is N-projective if, N N, any homomorphism ϕ : M N/N can be lifted to a homomorphism ϕ : M N. A module P is called projective if P is N-projective for all every R-module N. Lemma 1.3.24. If M i is N-projective, for i = 1,..., n (n N), then n i=1 M i is N-projective. If N is M j -projective, for j = 1,..., n, then N is n j=1 M j-projective. Lemma 1.3.25. (Lemma 4.47, [37]) Let M and N be right R-modules such that M is N-projective. If U = M N = L + N, then there exists M L such that U = M N. Proof. Let ι M and ι N be the natural inclusion homomorphism from M and N to U, respectively, and ϕ : U U/L be the natural epimorphism. Then ϕι N is an epimorphism. Since M is N-projective, there exists a homomorphism ϕ : M N such that ϕι N ϕ = ϕι M. We can construct the following submodule of L: M = {x ϕ (x) x M}. It is easy to check that M L. Since M N = 0 because x ϕ (x) N x N M x = 0, M M + N U = M N = M + N = M N. 16

Proposition 1.3.26. (Proposition 2, [18]) An abelian ring R is right Rickart if and only if R is left Rickart. Proof. It is enough to show that if ab = 0 then ba = 0. Let ab = 0. Since r R (a) = er for some e 2 = e R as R is right Rickart, b r R (a) = er b = eb = be as R is abelian. Thus, ba = bae = 0. Note that an abelian Rickart module is reduced: If a 2 = 0 then a r R (a) = er for some e 2 = e R. So, a = ea = ae = 0. Theorem 1.3.27. (Corollary 3.2, [49]) The following conditions are equivalent for a module M and S = End R (M): (a) S is a von Neumann regular ring; (b) for each ϕ S, Kerϕ and Imϕ are direct summands of M. Proof. (a) (b) Assume that S is a von Neumann regular ring. Then for any 0 ϕ S, there exists 0 e 2 = e S such that Sϕ = Se. Thus, Kerϕ = r M (Sϕ) = r M (Se) = (1 e)m. Also, we have ϕs = fs for some f 2 = f S. So, Imϕ = ϕs(m) = fs(m) = fm M. (b) (a) Let 0 ϕ S be arbitrary. By hypothesis, there exists a submodule N such that M = Kerϕ N. Since ϕm = ϕn M, there exists 0 ψ S such that ψϕ N = 1 N. Then (ϕ ϕψϕ)(m) = (ϕ ϕψϕ)(kerϕ N) = (ϕ ϕψϕ)(n) = 0. Hence ϕ ϕψϕ = 0. Thus S is a von Neumann regular ring. Theorem 1.3.28. (Theorem 2.1, [15]) A ring R is Baer and cononsingular if and only if R is extending and nonsingular. 17

Theorem 1.3.29. (Theorem 2.12, [40]) A module M is Baer and K-cononsingular if and only if M is extending and K-nonsingular. Proof. Suppose M is Baer and K-cononsingular. Since Kerϕ M for all ϕ S = End R (M), it is clear that M is K-nonsingular. Let L be a nonzero submodule of M. Since l S (L) = Se for some 1 e 2 = e S, L r M (l S (L)) = (1 e)m. Assume that L ess (1 e)m. Since em L ess M, by K-cononsingularity of M there exists a nonzero homomorphism ϕ S such that ϕ(em L) = 0. Thus, ϕ S(1 e) Se = 0, a contradiction. Hence L ess (1 e)m. Therefore M is extending. Conversely, suppose L is a submodule of M such that L ess M. Since M is extending there exists an idempotent 1 e S such that L ess em. Thus, (1 e)l = 0. Therefore M is K-cononsingular. Let N be a submodule of M. Then there exists an idempotent f S such that N ess fm. Hence l S (N) S(1 f). Assume that 0 ϕ l S (N) such that ϕ / S(1 f). We can safely assume that ϕ Sf, i.e., ϕ = ϕf. Since N (1 f)m ess M and ϕ(n (1 f)m) = 0, by K-nonsingularity of M ϕ = 0, a contradiction. Thus, l S (N) = S(1 f). Therefore M is Baer. Proposition 1.3.30. (Theorem 2.5, [43]) M is a Baer module if and only if End R (M) is a Baer ring and M is quasi-retractable. Theorem 1.3.31. The following equivalences hold true: (i) A module M is dual Baer iff ϕ J Imϕ M for all J End R (M) iff M has the SSSP and Imϕ M for all ϕ End R (M) (Theorem 2.1, [48]). (ii) R R is dual Baer iff R is a semisimple artinian ring (Corollary 2.9, [48]). 18

Proof. For the first equivalence of (i), let A S and N = ϕ A Imϕ. Then {ϕ S Imϕ N} = es for some e 2 = e S as M is dual Baer. Thus em N. For the other inclusion, let ϕ S be arbitrary such that Imϕ N. Then ϕ es, thus ϕm = eϕm em. Conversely, let N be a submodule of M. Consider D(N) = {ϕ S Imϕ N}. Since ϕ D(N) Imϕ = em for some e2 = e S, e D(N). Note that es D(N) as D(N) is a right ideal of S. For the other inclusion, let ϕ D(N) be arbitrary, hence Imϕ em. Thus ϕ = eϕ es. The second equivalence is trivial. For (ii), suppose R R is dual Baer. Let J be any subset of R. Then ϕ J ar. For any a R, consider the R-endomorphism f a End R (R) defined by f a (r) = ar where r R. Then for all a R, ar R R as R R is dual Baer. From (i), J R R. Therefore R is a semisimple artinian ring. The converse is obvious. 19

Chapter 2 RICKART MODULES 2.1 Definition and Properties of Rickart Modules In this section we introduce the notion of a Rickart module and obtain its basic properties. It is shown that every direct summand of a Rickart module inherits the property and that every Rickart module is K-nonsingular. Further, it satisfies D 2 condition and the Summand Intersection Property (SIP). In 2004, Rizvi and Roman [40] introduced the notion of Baer modules (see Definition 1.3.6) by exploiting the connections between a module M and its endomorphism ring S = End R (M). This notion puts the concept of Baer rings in the general module theoretic setting and has been investigated in recent years in [40], [41], [42], [43]. Recall that a ring R is said to be right Rickart if a R, r R (a) = er for some e 2 = e R. A significant amount of work has been done on right Rickart rings in the literature. However, not much is known about the Rickart property in the general module theoretic setting. Motivated by the concept of right Rickart rings and the notion of Baer modules, we define a Rickart module as follows: Definition 2.1.1. Let M be a right R-module and let S = End R (M). Then M is called a Rickart module if the right annihilator in M of any single element of S is 20

generated by an idempotent of S. Equivalently, ϕ S, r M (ϕ) = Kerϕ = em for some e 2 = e S. Note that Kerϕ = r M (ϕ) = r M (Sϕ) for ϕ S. Unlike the case of Baer rings and Baer modules, there is no symmetry available for the case of Rickart modules, just as a right (left) Rickart ring need not be left (right) Rickart. Also, we note that every Baer module is Rickart from the definition. In the following, we provide several other examples of Rickart modules including examples of Rickart modules which are not Baer: (Examples 2.1.17, 2.1.18 and 2.3.1, and Proposition 2.5.25 provide more instances of Rickart modules which are not Baer). Example 2.1.2. Every semisimple module is a Rickart module. R R is a Rickart module if R is a right Rickart ring: In particular, if R is a Baer ring, a von Neumann regular ring, or a right hereditary ring then R R is Rickart. Every Baer module is a Rickart module: For example, every nonsingular injective (or extending) module is Rickart (see Theorem 2.12, [40]). Every projective right R-module over a right hereditary ring R is a Rickart module (see Theorem 2.5.20). The free Z-module Z (I), for any index set I, is Rickart, while Z (I) is not a Baer Z-module if I is uncountable (see Remark 2.5.24). In particular, Z (N) ( = Z[x]) is a Rickart (and Baer) Z-module, while Z (R) is a Rickart but not a Baer Z-module. In general, if R is a right hereditary ring which is not Baer then every free R-module is Rickart but not Baer (Proposition 2.5.25). On the contrary, Z p and Z 4 are injective and quasi-injective Z-modules, respectively, neither of which is a Rickart Z-module. Next, we present a characterization of Rickart modules (see also Proposition 2.1.11, Theorem 2.1.19, Proposition 2.1.26 and Theorem 2.2.10). Proposition 2.1.3. The following conditions are equivalent for a module M: 21

(a) M is a Rickart module; (b) ϕ splits in M for any ϕ End R (M). Proof. Consider the short exact sequence, for ϕ End R (M) 0 Kerϕ = r M (ϕ) M ϕm 0. The short exact sequence splits in M iff Kerϕ M, i.e., iff M is Rickart. It is clear that Z Z is a Rickart module (Z is a domain) and its submodule 4Z Z is also a Rickart module. However, the Rickart property does not always transfer from a module to each of its submodules or conversely as the next two examples illustrate. Example 2.1.4. Every submodule of a Rickart module is not Rickart, in general: The Z-module Q Z 2 is Baer by Proposition 3.20 in [43], so it is Rickart. However, the submodule Z Z 2 is not a Rickart Z-module, even though Z and Z 2 are both Rickart Z-modules: The map (m, n) (0, m) has the kernel 2Z Z 2 Z Z 2, since Z Z is uniform. Example 2.1.5. Z 4 is not a Rickart Z-module: Let ϕ End Z (Z 4 ) such that ϕ(1) = 2. Since Kerϕ Z 4, Z 4 is not Rickart. However, the submodule 2Z 4 of Z 4 is a Rickart Z-module because 2Z 4 =Z Z 2. On the other hand, a direct summand of a Rickart module inherits the property. Theorem 2.1.6. Every direct summand of a Rickart module is a Rickart module. Proof. Let M be a Rickart module, N = em for some e 2 = e End R (M) and ψ End R (em). Then Kerψe = [em Kerψ] (1 e)m = Kerψ (1 e)m because for any m Kerψe, em em Kerψ, and hence m = em + (1 e)m 22

[em Kerψ] (1 e)m. The other inclusion follows easily. Since M is Rickart and ψe End R (M), Kerψe M and hence Kerψ M. Therefore Kerψ N, i.e., N is Rickart. Corollary 2.1.7. If R is a right Rickart ring then er, e 2 = e, is a Rickart R-module. Proof. Take M = R R in Theorem 2.1.6. Remark 2.1.8. Corollary 2.1.7 provides an answer to the motivating question raised in Section 1.1 (page 4). Further, if M is a Rickart module then so are Kerϕ and Imϕ for every ϕ End R (M) by Theorem 2.1.6. The following example shows that the converse of Theorem 2.1.6 is not true, in general. The example further illustrates that the direct sum of two Rickart modules need not be Rickart (see also Example 2.1.4). Example 2.1.9. Consider the ring R = ( Z 0 Z Z ) and the idempotent e=( 1 0 0 0 ). Let M = R R. Then M is not a Rickart module, while er = ( Z 0 Z 0 ) is a Rickart R-module: Note that End R (er) = ( Z 0 0 0 ). Then Ker ( u 0 0 0 ) = ( 0 0 0 0 ) for all 0 u Z showing er is Rickart. However, M is not Rickart: Consider ( 2 1 0 0 ) End R (M) = R. Then r M (( 2 0 1 0 )) = ( 0 1 ) Z which is not a direct summand of M. 0 2 Furthermore, note that (1 e)r = ( 0 0 0 Z ) = End R ((1 e)r). So, er and (1 e)r are Rickart R-modules, while their direct sum R R is not so. The question about when direct sums of Rickart modules are also Rickart will be studied later. While every submodule of a Rickart module is not Rickart by Example 2.1.4, our next result shows that the Rickart property for modules is inherited by fully invariant submodules under some conditions. 23

Proposition 2.1.10. Let M be a Rickart module and let N be a fully invariant submodule of M. If every endomorphism ϕ End R (N) can be extended to an endomorphism ϕ End R (M) then N is a Rickart module. Proof. Let ϕ End R (N) be arbitrary. By assumption, there exists ϕ End R (M) such that ϕ N = ϕ. Since M is Rickart, Kerϕ = em for some e = e 2 End R (M). Since ϕ(em) = 0, ϕ(en) = 0 en Kerϕ N = Kerϕ. Therefore Kerϕ = en. Since en N as N is a fully invariant submodule of M, N is a Rickart module. Note that if either N or M is M-injective, then every ϕ End R (N) can be extended to ϕ End R (M). Recall that a module M is said to have D 2 condition if N M with M/N = M M, we have N M. Proposition 2.1.11. A module M is Rickart if and only if M has D 2 condition and for all ϕ End R (M), Imϕ is isomorphic to a direct summand of M. Proof. Suppose M is a Rickart module. For any submodule N of M with M/N = M M, there exists ψ End R (M) such that Kerψ = N. Since M is Rickart, N M. The last statement is easy to check. The converse follows easily. Note that a module M is called w-d-rickart if Imϕ is isomorphic to a direct summand of M for all ϕ End R (M). Since every right Rickart ring is right nonsingular, we expect that a Rickart module will also have some kind of nonsingularity. Rizvi and Roman introduced the notion of K-nonsingularity and showed that every Baer module is K-nonsingular (Lemma 2.15, [40]). A module M is said to be K-nonsingular if, for all ϕ End R (M), r M (ϕ) = 24

Kerϕ ess M implies ϕ = 0. It is known that M is nonsingular M is polyform M is K-nonsingular, while the reverse implications are not true in general [42]. It was shown in Lemma 2.14 of [40] that a K-nonsingular, extending module is always a Baer module. Now, we show that every Rickart module is also K-nonsingular. Proposition 2.1.12. Every Rickart module is K-nonsingular. Proof. Let M be a Rickart module. Consider Kerϕ ess M for any ϕ End R (M). Since Kerϕ M, Kerϕ = M ϕ = 0. Remark 2.1.13. In view of Proposition 2.1.12 and the comment above it, every extending, Rickart module is Baer. The next reformulated proposition characterizes Baer modules in terms of the SSIP and the Rickart property for modules. Proposition 2.1.14. (Proposition 2.22, [40]) M is a Baer module if and only if M is a Rickart module with the SSIP. While every Baer module satisfies the SSIP by Proposition 2.1.14, for the case of Rickart modules only the weaker version, i.e., the SIP holds. Proposition 2.1.15. Every Rickart module has the SIP. Proof. Let M be a Rickart module and let L = em and N = fm for some nonzero idempotents e, f End R (M). Then Ker(1 f)e = [em Ker(1 f)] (1 e)m: For any y Ker(1 f)e, (1 f)(ey) = 0 and hence ey em Ker(1 f), then y = ey + (1 e)y [em Ker(1 f)] (1 e)m. The other inclusion is obvious. Since M is Rickart, Ker(1 f)e M. Therefore L N = em Ker(1 f) is a direct summand of M. 25

The following example shows that the converse of Proposition 2.1.15 is not true, in general. Example 2.1.16. Consider the Z-module Z p (where p is a prime number). Since Z p is indecomposable, it has the SIP. However Z p is not a Rickart module: Let ϕ : Z p Z p defined by ϕ(a) = ap. Since 0 Kerϕ Z p, Z p is not a Rickart Z-module. Similarly, the Z-module Z 4 satisfies the SIP as it is indecomposable but Z 4 is not a Rickart Z-module. In general, Rickart modules do not satisfy the SSIP, while Baer modules do. If a module M is Rickart but not Baer then M has the SIP by Proposition 2.1.15 but M does not satisfy the SSIP (see Proposition 2.1.14). The following examples due to Birkenmeier, Kim and Park exhibit Rickart modules which are not Baer modules. Example 2.1.17. (Example 1.7, [8]) Let A = n=1 Z 2. Consider T = {(a n ) n=1 A a n is eventually constant}, I = {(a n ) n=1 A a n = 0 eventually} = n=1 Z 2. ( ) Now, consider the ring R = T T/I 0 T/I and the idempotent e = ( ) (1,1,... ) 0+I 0 0+I in R. Note that R is a right hereditary ring, but R is not a Baer ring. Since R is a right Rickart ring (being right hereditary), M = er is a Rickart R-module by Theorem 2.1.6. However, M is not Baer. (See Proposition 1.3.30 and Example 2.2.15 in Section 3.) Example 2.1.18. (Example 1.6, [10]) For a field F, take F n = F for n = 1, 2, and let R = n=1 n=1 F n F n F n n=1 F n, 1 which is a subring of the 2 2 matrix ring over the ring n=1 F n, where n=1 F n, 1 is the F -algebra generated by n=1 F n and 1. Then R is a von Neumann regular 26 n=1

ring which is not a Baer ring. Denote the idempotent e = ( 0 0 0 1 ). Then M = er is a Rickart R-module by Theorem 2.1.6. However, M is not a Baer R-module because End R (M) = n=1 F n, 1 is not a Baer ring (see Proposition 1.3.30). It is obvious to note that in each of the above examples R R is also a Rickart module which is not Baer. As noted in the definition, in contrast to Baer modules, Rickart modules allow for the right annihilator in M of any single element of S to be generated by an idempotent in S = End R (M). However, this can be extended to the right annihilator of any nonempty finite subset of S, i.e., r M (I) = em, e 2 = e S for any nonempty finite subset I S as shown in the next result (note that r M (I) = r M (SI)). Theorem 2.1.19. The following conditions are equivalent for a module M: (a) M is a Rickart module; (b) the right annihilator in M of any finitely generated left ideal I = ϕ 1,, ϕ n of End R (M) is generated by an idempotent in End R (M). Proof. (a) (b) Suppose M is a Rickart module. Let I S S = End R (M) be any nonzero left ideal with a finite number of generators ϕ 1, ϕ 2,, ϕ n. Since r M (I) = n i=1 r M(ϕ i ) and r M (ϕ i ) M for 1 i n, r M (I) M as M has the SIP. (b) (a) Since for any ψ S, Sψ is a left ideal with one generator, r M (Sψ) M. Therefore M is a Rickart module. Definition 2.1.20. (Definition 1.3, [43]) A module M is called N-Rickart (or relatively Rickart to N) if, for every homomorphism ϕ : M N, Kerϕ M. In view of the above definition, a right R-module M is Rickart iff M is M-Rickart. 27

Example 2.1.21. Let M be a semisimple R-module. Then M is N-Rickart for any right R-module N. So, the simple Z-module Z p is Z-Rickart, but Z is not Z p -Rickart even though Z and Z p are Rickart Z-modules (where p is a prime number in N). Example 2.1.22. Z 4 is Z 3 -Rickart because there is no nonzero homomorphism from Z 4 to Z 3, but Z 4 is not a Rickart Z-module (see Example 2.1.5). We reformulate Proposition 1(b) of [51] in terms of our setting of Rickart modules. Proposition 2.1.23. Let a module M have the SIP and M i M j M. Then M i is M j -Rickart. Proof. Let ϕ Hom R (M i, M j ) and let N = {(m i, ϕm i ) m i M i }. Note that N M. Then Kerϕ = M i N M. Thus, Kerϕ M i. The next corollary is a direct consequence of Proposition 2.1.23. Corollary 2.1.24. If M M has the SIP then M is a Rickart module. Our next characterization extends Theorem 2.1.6. Theorem 2.1.25. Let M and N be right R-modules. Then M is N-Rickart if and only if for any direct summand M M and any submodule N N, M is N - Rickart. Proof. Let M = em for some e 2 = e End R (M), N N and ψ Hom R (M, N ) be arbitrary. Since ψem = ψm N N, Kerψe M. Since Kerψe = (1 e)m (Kerψe em) = (1 e)m Kerψ, Kerψ M Kerψ M. Thus M is N -Rickart. The converse follows easily. 28