Abelian groups whose endomorphism rings are von Neumann regular have been extensively investigate... more Abelian groups whose endomorphism rings are von Neumann regular have been extensively investigated in the literature. In this paper, we study modules whose endomorphism rings are von Neumann regular, which we call endoregular modules. We provide characterizations of endoregular modules and investigate their properties. Some classes of rings R are characterized in terms of endoregular R-modules. It is shown that a direct summand of an endoregular module inherits the property, while a direct sum of endoregular modules does not. Necessary and sufficient conditions for a finite direct sum of endoregular modules to be an endoregular module are provided. We show that EndR(M) is strongly regular precisely when a module M decomposes into a direct sum of the image and the kernel of any ϕ ∈ EndR(M). As a special case, modules whose endomorphism rings are semisimple artinian are characterized. We provide a precise description of an indecomposable endoregular module over an arbitrary commutative ring. A structure theorem for extending endoregular abelian groups is also provided.
We investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under v... more We investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under various ring extensions including group ring extensions, full and triangular matrix ring extensions, and infinite matrix ring extensions. As a consequence, we show that for semiprime rings R and S, if R and S are Morita equivalent, then so are the quasi-Baer right ring hulls of R and S, respectively. As an application, we prove that if unital C*-algebras A and B are Morita equivalent as rings, then the bounded central closure of A and that of B are strongly Morita equivalent as C*-algebras. Our results show that the quasi-Baer property is always preserved by infinite matrix rings, unlike the Baer property. Moreover, we give an affirmative answer to an open question of Goel and Jain for the commutative group ring A[G] of a torsion-free Abelian group G over a commutative semiprime quasi-continuous ring A. Examples that illustrate and delimit the results of this paper are provided.
Ring theory. Proceedings of the biennial Ohio State-Denison mathematics conference, May 14-16, 1992 dedicated to the memory of H. J. Zassenhaus
A characterization of semiperfect rings and modules, G. Azumaya on universal localizations at sem... more A characterization of semiperfect rings and modules, G. Azumaya on universal localizations at semiprime Goldie ideals, J. Beachy strongly and properly semiprime modules and rings, K. Beidar and R. Wisbauer enveloping algebras of infinite dimensional lie algebras and lie superalgebras, J. Bergen primes of skew fields, H.H. Brungs uniform ranks of prime factors of skew polynomial rings, K.R. Goodearl weakly-projective modules, S.K. Jain, et al classification of modules suitable for transferring properties between the modules and their endomorphism rings, S.M. Khuri ring theoretic properties of the co-ordinate rings of quantum symplectic and euclidean space, I.M. Musson when direct sums of singular injectives are injective, S.S. Page and Y. Zhou polynomials over division rings, and their applications, L. Rowen characterization of rings using lifting and extending modules, N. Vanaja right locally distributive rings, M.I. Wright on weakly semi-primitive rings, J.M. Zelmanowitz a convenient source of homological examples over artinian rings, B. Zimmermann-Huisgen. Part contents.
A module M is called continuous if (i) every submodule of M is essential in a summand of M, and (... more A module M is called continuous if (i) every submodule of M is essential in a summand of M, and (ii) if a submodule A is isomorphic to a summand of M, then A is itself a summand of M. Injective and quasi-injective modules play an important role in module theory and continuous modules are a generalization of these concepts. Many of the important properties that hold for (quasi-) injective modules, still hold for continuous modules, and it is often more convenient to work with the above two conditions rather than the notion of (quasi-) injectivity. This thesis deals with several important aspects of the theory of continuous modules. We give a decomposition theorem for continuous modules and, as a corollary, obtain a partial generalization of a result of Matlis and Papp. We also answer the open question: When is a finite direct sum of indecomposable continous modules continuous modules is also examined. The main chapter deals with the concept of continuous hulls. We give an appropriate...
1. Introduction. This paper studies direct sums M=(B iζΞI M i of indecomposable modules. Specific... more 1. Introduction. This paper studies direct sums M=(B iζΞI M i of indecomposable modules. Specifically, we give a number of necessary and sufficient conditions for such a sum to be quasi-continuous or continuous. This question was settled in [6], in a very satisfactory way, in case the index set / is finite, or the ring is right-noetherian, but the general case dealt with here is much more complicated. Such sums M=@ i^I M i have been investigated in great detail, in a long series of papers since about 1970, by M. Harada and his collaborators, usually under the additional hypothesis that the M { have local endomorphism rings (so that the Krull-Schmidt-Azumaya Theorem applies). One of the central results is the following: Theorem 1 ([3], p. 22). For a module with a decomposition M=® ieI M h and with all endoίM,) local, the following statements are equivalent: (1) The decomposition is locally semi-T-nilpotent; (2) it complements direct summands (3) any local direct summand of M is a direct summand. (The relevent terms are defined later on in this section.
We show the existence of principally (and finitely generated) right FI-extending right ring hulls... more We show the existence of principally (and finitely generated) right FI-extending right ring hulls for semiprime rings. From this result, we prove that right principally quasi-Baer (i.e., right p.q.-Baer) right ring hulls always exist for semiprime rings. This existence of right p.q.-Baer right ring hull for a semiprime ring unifies the result by Burgess and Raphael on the existence of
Abelian groups whose endomorphism rings are von Neumann regular have been extensively investigate... more Abelian groups whose endomorphism rings are von Neumann regular have been extensively investigated in the literature. In this paper, we study modules whose endomorphism rings are von Neumann regular, which we call endoregular modules. We provide characterizations of endoregular modules and investigate their properties. Some classes of rings R are characterized in terms of endoregular R-modules. It is shown that a direct summand of an endoregular module inherits the property, while a direct sum of endoregular modules does not. Necessary and sufficient conditions for a finite direct sum of endoregular modules to be an endoregular module are provided. We show that EndR(M) is strongly regular precisely when a module M decomposes into a direct sum of the image and the kernel of any ϕ ∈ EndR(M). As a special case, modules whose endomorphism rings are semisimple artinian are characterized. We provide a precise description of an indecomposable endoregular module over an arbitrary commutative ring. A structure theorem for extending endoregular abelian groups is also provided.
We investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under v... more We investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under various ring extensions including group ring extensions, full and triangular matrix ring extensions, and infinite matrix ring extensions. As a consequence, we show that for semiprime rings R and S, if R and S are Morita equivalent, then so are the quasi-Baer right ring hulls of R and S, respectively. As an application, we prove that if unital C*-algebras A and B are Morita equivalent as rings, then the bounded central closure of A and that of B are strongly Morita equivalent as C*-algebras. Our results show that the quasi-Baer property is always preserved by infinite matrix rings, unlike the Baer property. Moreover, we give an affirmative answer to an open question of Goel and Jain for the commutative group ring A[G] of a torsion-free Abelian group G over a commutative semiprime quasi-continuous ring A. Examples that illustrate and delimit the results of this paper are provided.
Ring theory. Proceedings of the biennial Ohio State-Denison mathematics conference, May 14-16, 1992 dedicated to the memory of H. J. Zassenhaus
A characterization of semiperfect rings and modules, G. Azumaya on universal localizations at sem... more A characterization of semiperfect rings and modules, G. Azumaya on universal localizations at semiprime Goldie ideals, J. Beachy strongly and properly semiprime modules and rings, K. Beidar and R. Wisbauer enveloping algebras of infinite dimensional lie algebras and lie superalgebras, J. Bergen primes of skew fields, H.H. Brungs uniform ranks of prime factors of skew polynomial rings, K.R. Goodearl weakly-projective modules, S.K. Jain, et al classification of modules suitable for transferring properties between the modules and their endomorphism rings, S.M. Khuri ring theoretic properties of the co-ordinate rings of quantum symplectic and euclidean space, I.M. Musson when direct sums of singular injectives are injective, S.S. Page and Y. Zhou polynomials over division rings, and their applications, L. Rowen characterization of rings using lifting and extending modules, N. Vanaja right locally distributive rings, M.I. Wright on weakly semi-primitive rings, J.M. Zelmanowitz a convenient source of homological examples over artinian rings, B. Zimmermann-Huisgen. Part contents.
A module M is called continuous if (i) every submodule of M is essential in a summand of M, and (... more A module M is called continuous if (i) every submodule of M is essential in a summand of M, and (ii) if a submodule A is isomorphic to a summand of M, then A is itself a summand of M. Injective and quasi-injective modules play an important role in module theory and continuous modules are a generalization of these concepts. Many of the important properties that hold for (quasi-) injective modules, still hold for continuous modules, and it is often more convenient to work with the above two conditions rather than the notion of (quasi-) injectivity. This thesis deals with several important aspects of the theory of continuous modules. We give a decomposition theorem for continuous modules and, as a corollary, obtain a partial generalization of a result of Matlis and Papp. We also answer the open question: When is a finite direct sum of indecomposable continous modules continuous modules is also examined. The main chapter deals with the concept of continuous hulls. We give an appropriate...
1. Introduction. This paper studies direct sums M=(B iζΞI M i of indecomposable modules. Specific... more 1. Introduction. This paper studies direct sums M=(B iζΞI M i of indecomposable modules. Specifically, we give a number of necessary and sufficient conditions for such a sum to be quasi-continuous or continuous. This question was settled in [6], in a very satisfactory way, in case the index set / is finite, or the ring is right-noetherian, but the general case dealt with here is much more complicated. Such sums M=@ i^I M i have been investigated in great detail, in a long series of papers since about 1970, by M. Harada and his collaborators, usually under the additional hypothesis that the M { have local endomorphism rings (so that the Krull-Schmidt-Azumaya Theorem applies). One of the central results is the following: Theorem 1 ([3], p. 22). For a module with a decomposition M=® ieI M h and with all endoίM,) local, the following statements are equivalent: (1) The decomposition is locally semi-T-nilpotent; (2) it complements direct summands (3) any local direct summand of M is a direct summand. (The relevent terms are defined later on in this section.
We show the existence of principally (and finitely generated) right FI-extending right ring hulls... more We show the existence of principally (and finitely generated) right FI-extending right ring hulls for semiprime rings. From this result, we prove that right principally quasi-Baer (i.e., right p.q.-Baer) right ring hulls always exist for semiprime rings. This existence of right p.q.-Baer right ring hull for a semiprime ring unifies the result by Burgess and Raphael on the existence of
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