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.
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...
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
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.
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...
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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