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Jae Keol Park - One of the best experts on this subject based on the ideXlab platform.
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Quasi-Baer module hulls and applications
Journal of Pure and Applied Algebra, 2018Co-Authors: Gangyong Lee, Jae Keol Park, S. Tariq Rizvi, Cosmin S. RomanAbstract:Abstract Let V be a module with S = End ( V ) . V is called a quasi-Baer module if for each ideal J of S , r V ( J ) = e V for some e 2 = e ∈ S . On the other hand, V is called a Rickart module if for each ϕ ∈ S , Ker ( ϕ ) = e V for some e 2 = e ∈ S . For a module N , the quasi-Baer module hull qB ( N ) (resp., the Rickart module hull Ric ( N ) ) of N , if it exists, is the smallest quasi-Baer (resp., Rickart) overmodule, in a fixed injective hull E ( N ) of N . In this paper, we initiate the study of quasi-Baer and Rickart module hulls. When a ring R is semiprime and ideal intrinsic over its center, it is shown that every finitely generated projective R -module has a quasi-Baer hull. Let R be a Dedekind domain with F its field of fractions and let { K i | i ∈ Λ } be any set of R -submodules of F R . For an R -module M R with Ann R ( M ) ≠ 0 , we show that M R ⊕ ( ⨁ i ∈ Λ K i ) R has a quasi-Baer module hull if and only if M R is semisimple. This quasi-Baer hull is explicitly described. An example such that M R ⊕ ( ⨁ i ∈ Λ K i ) R has no Rickart module hull is constructed. If N is a module over a Dedekind domain for which N / t ( N ) is projective and Ann R ( t ( N ) ) ≠ 0 , where t ( N ) is the torsion submodule of N , we show that the quasi-Baer hull qB ( N ) of N exists if and only if t ( N ) is semisimple. We prove that the Rickart module hull also exists for such modules N . Furthermore, we provide explicit constructions of qB ( N ) and Ric ( N ) and show that in this situation these two hulls coincide. Among applications, it is shown that if N is a finitely generated module over a Dedekind domain, then N is quasi-Baer if and only if N is Rickart if and only if N is Baer if and only if N is semisimple or torsion-free. For a direct sum N R of finitely generated modules, where R is a Dedekind domain, we show that N is quasi-Baer if and only if N is Rickart if and only if N is semisimple or torsion-free. Examples exhibiting differences between the notions of a Baer hull, a quasi-Baer hull, and a Rickart hull of a module are presented. Various explicit examples illustrating our results are constructed.
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Baer module hulls of certain modules over a Dedekind domain
Journal of Algebra and Its Applications, 2016Co-Authors: Jae Keol Park, S. Tariq RizviAbstract:The notion of a Baer ring, introduced by Kaplansky, has been extended to that of a Baer module using the endomorphism ring of a module in recent years. There do exist some results in the literature on Baer ring hulls of given rings. In contrast, the study of a Baer module hull of a given module remains wide open. In this paper we initiate this study. For a given module [Formula: see text], the Baer module hull, [Formula: see text], is the smallest Baer overmodule contained in a fixed injective hull [Formula: see text] of [Formula: see text]. For a certain class of modules [Formula: see text] over a commutative Noetherian domain, we characterize all essential overmodules of [Formula: see text] which are Baer. As a consequence, it is shown that Baer module hulls exist for such modules over a Dedekind domain. A precise description of such hulls is obtained. It is proved that a finitely generated module [Formula: see text] over a Dedekind domain has a Baer module hull if and only if the torsion submodule [Formula: see text] of [Formula: see text] is semisimple. Further, in this case, the Baer module hull of [Formula: see text] is explicitly described. As applications, various properties and examples of Baer hulls are exhibited. It is shown that if [Formula: see text] are two modules with Baer hulls, [Formula: see text] may not have a Baer hull. On the other hand, the Baer module hull of the [Formula: see text]-module [Formula: see text] ([Formula: see text] a prime integer) is precisely given by [Formula: see text]. It is shown that infinitely generated modules over a Dedekind domain may not have Baer module hulls.
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Baer module hulls of certain modules over a Dedekind domain
Journal of Algebra and Its Applications, 2016Co-Authors: Jae Keol Park, S. Tariq RizviAbstract:The notion of a Baer ring, introduced by Kaplansky, has been extended to that of a Baer module using the endomorphism ring of a module in recent years. There do exist some results in the literature on Baer ring hulls of given rings. In contrast, the study of a Baer module hull of a given module remains wide open. In this paper we initiate this study. For a given module M, the Baer module hull, 𝔅(M), is the smallest Baer overmodule contained in a fixed injective hull E(M) of M. For a certain class of modules N over a commutative Noetherian domain, we characterize all essential overmodules of N which are Baer. As a consequence, it is shown that Baer module hulls exist for such modules over a Dedekind domain. A precise description of such hulls is obtained. It is proved that a finitely generated module N over a Dedekind domain has a Baer module hull if and only if the torsion submodule t(N) of N is semisimple. Further, in this case, the Baer module hull of N is explicitly described. As applications, various ...
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Baer, Quasi-Baer Modules, and Their Applications
Extensions of Rings and Modules, 2013Co-Authors: Gary F. Birkenmeier, Jae Keol Park, S. Tariq RizviAbstract:The Baer and the quasi-Baer properties of rings are extended to a module theoretic setting in this chapter. Using the endomorphism ring of a module, the notions of Baer, quasi-Baer, and Rickart modules are introduced and studied. Similar to the fact that every Baer ring is nonsingular, we shall see that every Baer module satisfies a weaker notion of nonsingularity of modules (\(\mathcal{K}\)-nonsingularity) which depends on the endomorphism ring of the module. Strong connections between a Baer module and an extending module will be observed via this weak nonsingularity and its dual notion. It is shown that an extending module which is \(\mathcal{K}\)-nonsingular is precisely a \(\mathcal{K}\)-cononsingular Baer module. This provides a module theoretic analogue of the Chatters-Khuri theorem for rings. Direct summands of Baer and quasi-Baer modules respectively inherit these properties. This provides a rich source of examples of Baer and quasi-Baer modules, since one can readily see that for any (quasi-)Baer ring R and an idempotent e in R, the right R-module eR R is always a (quasi-)Baer module. It will be seen that every projective module over a quasi-Baer ring is a quasi-Baer module. Connections of a (quasi-)Baer module and its endomorphism ring are discussed. Characterizations of classes of rings via the Baer property of certain classes of free modules over them are presented. An application also yields a type theory for \(\mathcal{K}\)-nonsingular extending (continuous) modules which, in particular, improve the type theory for nonsingular injective modules provided by Goodearl and Boyle.
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Baer, Rickart, and Quasi-Baer Rings
Extensions of Rings and Modules, 2013Co-Authors: Gary F. Birkenmeier, Jae Keol Park, S. Tariq RizviAbstract:This chapter is devoted to properties of rings for which certain annihilators are direct summands. Such classes of rings include those of Baer rings, right Rickart rings, quasi-Baer rings, and right p.q.-Baer rings. The results and the material presented in this chapter will be instrumental in developing the subject of our study in later chapters. It is shown that the Baer and the Rickart properties of rings do not transfer to the rings of matrices or to the polynomial ring extensions, while the quasi-Baer and the p.q.-Baer properties of rings do so. The notions of Baer and Rickart rings are compared and contrasted in Sect. 3.1 and the notions of quasi-Baer and principally quasi-Baer rings in Sect. 3.2, respectively. A result of Chatters and Khuri shows that there are strong bonds between the extending and the Baer properties of rings. We shall also see some instances where the two notions coincide. It is shown that there are close connections between the FI-extending and the quasi-Baer properties for rings.
Gary F. Birkenmeier - One of the best experts on this subject based on the ideXlab platform.
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π-Baer rings
Journal of Algebra and Its Applications, 2017Co-Authors: Gary F. Birkenmeier, Yeliz Kara, Adnan TercanAbstract:We say a ring R is π-Baer if the right annihilator of every projection invariant left ideal of R is generated by an idempotent element of R. In this paper, we study connections between the π-Baer condition and related conditions such as the Baer, quasi-Baer and π-extending conditions. The 2-by-2 generalized triangular and the n-by-n triangular π-Baer matrix rings are characterized. Also, we prove that a n-by-n full matrix ring over a π-Baer ring is a π-Baer ring. In contrast to the Baer condition, it is shown that the π-Baer condition transfers from a base ring to many of its polynomial extensions. Examples are provided to illustrate and delimit our results.
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Baer, Quasi-Baer Modules, and Their Applications
Extensions of Rings and Modules, 2013Co-Authors: Gary F. Birkenmeier, Jae Keol Park, S. Tariq RizviAbstract:The Baer and the quasi-Baer properties of rings are extended to a module theoretic setting in this chapter. Using the endomorphism ring of a module, the notions of Baer, quasi-Baer, and Rickart modules are introduced and studied. Similar to the fact that every Baer ring is nonsingular, we shall see that every Baer module satisfies a weaker notion of nonsingularity of modules (\(\mathcal{K}\)-nonsingularity) which depends on the endomorphism ring of the module. Strong connections between a Baer module and an extending module will be observed via this weak nonsingularity and its dual notion. It is shown that an extending module which is \(\mathcal{K}\)-nonsingular is precisely a \(\mathcal{K}\)-cononsingular Baer module. This provides a module theoretic analogue of the Chatters-Khuri theorem for rings. Direct summands of Baer and quasi-Baer modules respectively inherit these properties. This provides a rich source of examples of Baer and quasi-Baer modules, since one can readily see that for any (quasi-)Baer ring R and an idempotent e in R, the right R-module eR R is always a (quasi-)Baer module. It will be seen that every projective module over a quasi-Baer ring is a quasi-Baer module. Connections of a (quasi-)Baer module and its endomorphism ring are discussed. Characterizations of classes of rings via the Baer property of certain classes of free modules over them are presented. An application also yields a type theory for \(\mathcal{K}\)-nonsingular extending (continuous) modules which, in particular, improve the type theory for nonsingular injective modules provided by Goodearl and Boyle.
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Baer, Rickart, and Quasi-Baer Rings
Extensions of Rings and Modules, 2013Co-Authors: Gary F. Birkenmeier, Jae Keol Park, S. Tariq RizviAbstract:This chapter is devoted to properties of rings for which certain annihilators are direct summands. Such classes of rings include those of Baer rings, right Rickart rings, quasi-Baer rings, and right p.q.-Baer rings. The results and the material presented in this chapter will be instrumental in developing the subject of our study in later chapters. It is shown that the Baer and the Rickart properties of rings do not transfer to the rings of matrices or to the polynomial ring extensions, while the quasi-Baer and the p.q.-Baer properties of rings do so. The notions of Baer and Rickart rings are compared and contrasted in Sect. 3.1 and the notions of quasi-Baer and principally quasi-Baer rings in Sect. 3.2, respectively. A result of Chatters and Khuri shows that there are strong bonds between the extending and the Baer properties of rings. We shall also see some instances where the two notions coincide. It is shown that there are close connections between the FI-extending and the quasi-Baer properties for rings.
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THE FACTOR RING OF A QUASI-Baer RING BY ITS PRIME RADICAL
Journal of Algebra and Its Applications, 2011Co-Authors: Gary F. Birkenmeier, Jin Yong Kim, Jae Keol ParkAbstract:The quasi-Baer condition of R/P(R) is investigated when R is a quasi-Baer ring, where P(R) is the prime radical of R. We provide an example of quasi-Baer ring R such that R/P(R) is not quasi-Baer. However, when P(R) is nilpotent, we prove that if R is a quasi-Baer (resp., Baer) ring, then R/P(R) is quasi-Baer (resp., Baer). Examples which illustrate and delimit the results of this paper are provided.
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Principally Quasi-Baer Ring Hulls
Advances in Ring Theory, 2010Co-Authors: Gary F. Birkenmeier, Jae Keol Park, S. Tariq RizviAbstract: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 a closely related unique smallest overring for a von Neumann regular ring with bounded index and the result of Dobbs and Picavet showing the existence of a weak Baer envelope for a commutative semiprime ring. As applications, we illustrate the transference of certain properties between a semiprime ring and its right p.q.-Baer right ring hull, and we explicitly describe a structure theorem for the right p.q.-Baer right ring hull of a semiprime ring with only finitely many minimal prime ideals. The existence of PP right ring hulls for reduced rings is also obtained. Further application to ring extensions such as monoid rings, matrix, and triangular matrix rings are investigated. Moreover, examples and counterexamples are provided.
Ahmad Moussavi - One of the best experts on this subject based on the ideXlab platform.
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Polynomial extensions of modules with the quasi-Baer property
Journal of Algebra, 2020Co-Authors: P. Amirzadeh Dana, Ahmad MoussaviAbstract:Abstract In this paper it is shown that, for a module M over a ring R with S = E n d R ( M ) , the endomorphism ring of the R [ x ] -module M [ x ] is isomorphic to a subring of S [ [ x ] ] . Also the endomorphism ring of the R [ [ x ] ] -module M [ [ x ] ] is isomorphic to S [ [ x ] ] . As a consequence, we show that for a module M R and an arbitrary nonempty set of not necessarily commuting indeterminates X, M R is quasi-Baer if and only if M [ X ] R [ X ] is quasi-Baer if and only if M [ [ X ] ] R [ [ X ] ] is quasi-Baer if and only if M [ x ] R [ x ] is quasi-Baer if and only if M [ [ x ] ] R [ [ x ] ] is quasi-Baer. Moreover, a module M R with IFP, is Baer if and only if M [ x ] R [ x ] is Baer if and only if M [ [ x ] ] R [ [ x ] ] is Baer. It is also shown that, when M R is a finitely generated module, and every semicentral idempotent in S is central, then M [ [ X ] ] R [ [ X ] ] is endo-p.q.-Baer if and only if M [ [ x ] ] R [ [ x ] ] is endo-p.q.-Baer if and only if M R is endo-p.q.-Baer and every countable family of fully invariant direct summand of M has a generalized countable join. Our results extend several existing results.
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Endo-principally quasi-Baer modules
Journal of Algebra and Its Applications, 2015Co-Authors: P. Amirzadeh Dana, Ahmad MoussaviAbstract:Analogous to left p.q.-Baer property of a ring [G. F. Birkenmeier, J. Y. Kim and J. K. Park, Principally quasi-Baer rings, Comm. Algebra29 (2001) 639–660], we say a right R-module M is endo-principallyquasi-Baer (or simply, endo-p.q.-Baer) if for every m ∈ M, lS(Sm) = Se for some e2 = e ∈ S = EndR(M). It is shown that every direct summand of an endo-p.q.-Baer module inherits the property that any projective (free) module over a left p.q.-Baer ring is an endo-p.q.-Baer module. In particular, the endomorphism ring of every infinitely generated free right R-module is a left (or right) p.q.-Baer ring if and only if R is quasi-Baer. Furthermore, every principally right ℱℐ-extending right ℱℐ-𝒦-nonsingular ring is left p.q.-Baer and every left p.q.-Baer right ℱℐ-𝒦-cononsingular ring is principally right ℱℐ-extending.
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Weakly principally quasi-Baer rings
Journal of Algebra and Its Applications, 2015Co-Authors: A. Majidinya, Ahmad MoussaviAbstract:We say a ring R with unity is weakly principally quasi-Baer or simply (weakly p.q.-Baer) if the right annihilator of a principal right ideal is right s-unital by right semicentral idempotents, which implies that R modulo the right annihilator of any principal right ideal is flat. It is proven that the weakly p.q.-Baer property is inherited by polynomial extensions and includes both left p.q.-Baer rings and right p.q.-Baer rings and is closed under direct products and Morita invariance. A ring R is weakly p.q.-Baer if and only if the upper triangular matrix ring Tn(R) is weakly p.q.-Baer. We extend a theorem of Kist for commutative Baer rings to weakly p.q.-Baer rings for which every prime ideal contains a unique minimal prime ideal without using topological arguments. It is also shown that there is an important subclass of weakly p.q.-Baer rings which have a nontrivial subdirect product representation.
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On Skew Quasi-Baer Rings
Communications in Algebra, 2010Co-Authors: Mohammad Habibi, Ahmad Moussavi, R. ManaviyatAbstract:A ring R with an automorphism α and an α-derivation δ is called (α,δ)-quasi-Baer (resp., quasi-Baer) if the right annihilator of every (α,δ)-ideal (resp. ideal) of R is generated by an idempotent, as a right ideal. We show the left-right symmetry of (α, δ)-quasi Baer condition and prove that a ring R is (α, δ)-quasi Baer if and only if R[x; α, δ] is α-quasi Baer if and only if R[x; α, δ] is -quasi Baer for every extended derivation of δ. When R is a ring with IFP, then R is (α, δ)-Baer if and only if R[x; α, δ] is α-Baer if and only if R[x; α, δ] is -Baer for every extended α-derivation on R[x; α, δ] of δ. A rich source of examples for (α, δ)-quasi Baer rings is provided.
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Baer and Quasi-Baer Differential Polynomial Rings
Communications in Algebra, 2008Co-Authors: Alireza Nasr-isfahani, Ahmad MoussaviAbstract:A ring R with a derivation δ is called δ-quasi Baer (resp. quasi-Baer), if the right annihilator of every δ-ideal (resp. ideal) of R is generated by an idempotent, as a right ideal. We show the left-right symmetry of δ-(quasi) Baer condition and prove that a ring R is δ-quasi Baer if and only if R[x;δ] is quasi Baer if and only if R[x;δ] is -quasi Baer for every extended derivation of δ. When R is a ring with IFP, then R is δ-Baer if and only if R[x;δ] is Baer if and only if R[x;δ] is -Baer for every extended derivation of δ. A rich source of examples for δ-(quasi) Baer rings is provided.
S. Tariq Rizvi - One of the best experts on this subject based on the ideXlab platform.
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Quasi-Baer module hulls and applications
Journal of Pure and Applied Algebra, 2018Co-Authors: Gangyong Lee, Jae Keol Park, S. Tariq Rizvi, Cosmin S. RomanAbstract:Abstract Let V be a module with S = End ( V ) . V is called a quasi-Baer module if for each ideal J of S , r V ( J ) = e V for some e 2 = e ∈ S . On the other hand, V is called a Rickart module if for each ϕ ∈ S , Ker ( ϕ ) = e V for some e 2 = e ∈ S . For a module N , the quasi-Baer module hull qB ( N ) (resp., the Rickart module hull Ric ( N ) ) of N , if it exists, is the smallest quasi-Baer (resp., Rickart) overmodule, in a fixed injective hull E ( N ) of N . In this paper, we initiate the study of quasi-Baer and Rickart module hulls. When a ring R is semiprime and ideal intrinsic over its center, it is shown that every finitely generated projective R -module has a quasi-Baer hull. Let R be a Dedekind domain with F its field of fractions and let { K i | i ∈ Λ } be any set of R -submodules of F R . For an R -module M R with Ann R ( M ) ≠ 0 , we show that M R ⊕ ( ⨁ i ∈ Λ K i ) R has a quasi-Baer module hull if and only if M R is semisimple. This quasi-Baer hull is explicitly described. An example such that M R ⊕ ( ⨁ i ∈ Λ K i ) R has no Rickart module hull is constructed. If N is a module over a Dedekind domain for which N / t ( N ) is projective and Ann R ( t ( N ) ) ≠ 0 , where t ( N ) is the torsion submodule of N , we show that the quasi-Baer hull qB ( N ) of N exists if and only if t ( N ) is semisimple. We prove that the Rickart module hull also exists for such modules N . Furthermore, we provide explicit constructions of qB ( N ) and Ric ( N ) and show that in this situation these two hulls coincide. Among applications, it is shown that if N is a finitely generated module over a Dedekind domain, then N is quasi-Baer if and only if N is Rickart if and only if N is Baer if and only if N is semisimple or torsion-free. For a direct sum N R of finitely generated modules, where R is a Dedekind domain, we show that N is quasi-Baer if and only if N is Rickart if and only if N is semisimple or torsion-free. Examples exhibiting differences between the notions of a Baer hull, a quasi-Baer hull, and a Rickart hull of a module are presented. Various explicit examples illustrating our results are constructed.
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Baer module hulls of certain modules over a Dedekind domain
Journal of Algebra and Its Applications, 2016Co-Authors: Jae Keol Park, S. Tariq RizviAbstract:The notion of a Baer ring, introduced by Kaplansky, has been extended to that of a Baer module using the endomorphism ring of a module in recent years. There do exist some results in the literature on Baer ring hulls of given rings. In contrast, the study of a Baer module hull of a given module remains wide open. In this paper we initiate this study. For a given module [Formula: see text], the Baer module hull, [Formula: see text], is the smallest Baer overmodule contained in a fixed injective hull [Formula: see text] of [Formula: see text]. For a certain class of modules [Formula: see text] over a commutative Noetherian domain, we characterize all essential overmodules of [Formula: see text] which are Baer. As a consequence, it is shown that Baer module hulls exist for such modules over a Dedekind domain. A precise description of such hulls is obtained. It is proved that a finitely generated module [Formula: see text] over a Dedekind domain has a Baer module hull if and only if the torsion submodule [Formula: see text] of [Formula: see text] is semisimple. Further, in this case, the Baer module hull of [Formula: see text] is explicitly described. As applications, various properties and examples of Baer hulls are exhibited. It is shown that if [Formula: see text] are two modules with Baer hulls, [Formula: see text] may not have a Baer hull. On the other hand, the Baer module hull of the [Formula: see text]-module [Formula: see text] ([Formula: see text] a prime integer) is precisely given by [Formula: see text]. It is shown that infinitely generated modules over a Dedekind domain may not have Baer module hulls.
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Baer module hulls of certain modules over a Dedekind domain
Journal of Algebra and Its Applications, 2016Co-Authors: Jae Keol Park, S. Tariq RizviAbstract:The notion of a Baer ring, introduced by Kaplansky, has been extended to that of a Baer module using the endomorphism ring of a module in recent years. There do exist some results in the literature on Baer ring hulls of given rings. In contrast, the study of a Baer module hull of a given module remains wide open. In this paper we initiate this study. For a given module M, the Baer module hull, 𝔅(M), is the smallest Baer overmodule contained in a fixed injective hull E(M) of M. For a certain class of modules N over a commutative Noetherian domain, we characterize all essential overmodules of N which are Baer. As a consequence, it is shown that Baer module hulls exist for such modules over a Dedekind domain. A precise description of such hulls is obtained. It is proved that a finitely generated module N over a Dedekind domain has a Baer module hull if and only if the torsion submodule t(N) of N is semisimple. Further, in this case, the Baer module hull of N is explicitly described. As applications, various ...
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Direct sums of quasi-Baer modules
Journal of Algebra, 2016Co-Authors: Gangyong Lee, S. Tariq RizviAbstract:Abstract While the research work on quasi-Baer rings has been quite extensive in existing literature, the study of quasi-Baer modules introduced in 2004 remains quite limited with only little work available on the notion. In this paper, we extend the study of quasi-Baer modules. A characterization of a quasi-Baer module in terms of its endomorphism ring is obtained. We provide a complete characterization of arbitrary direct sums of quasi-Baer modules to be quasi-Baer. We also show characterizations of some important properties of quasi-Baer modules. Examples which delineate the concepts and the results are provided.
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Baer, Quasi-Baer Modules, and Their Applications
Extensions of Rings and Modules, 2013Co-Authors: Gary F. Birkenmeier, Jae Keol Park, S. Tariq RizviAbstract:The Baer and the quasi-Baer properties of rings are extended to a module theoretic setting in this chapter. Using the endomorphism ring of a module, the notions of Baer, quasi-Baer, and Rickart modules are introduced and studied. Similar to the fact that every Baer ring is nonsingular, we shall see that every Baer module satisfies a weaker notion of nonsingularity of modules (\(\mathcal{K}\)-nonsingularity) which depends on the endomorphism ring of the module. Strong connections between a Baer module and an extending module will be observed via this weak nonsingularity and its dual notion. It is shown that an extending module which is \(\mathcal{K}\)-nonsingular is precisely a \(\mathcal{K}\)-cononsingular Baer module. This provides a module theoretic analogue of the Chatters-Khuri theorem for rings. Direct summands of Baer and quasi-Baer modules respectively inherit these properties. This provides a rich source of examples of Baer and quasi-Baer modules, since one can readily see that for any (quasi-)Baer ring R and an idempotent e in R, the right R-module eR R is always a (quasi-)Baer module. It will be seen that every projective module over a quasi-Baer ring is a quasi-Baer module. Connections of a (quasi-)Baer module and its endomorphism ring are discussed. Characterizations of classes of rings via the Baer property of certain classes of free modules over them are presented. An application also yields a type theory for \(\mathcal{K}\)-nonsingular extending (continuous) modules which, in particular, improve the type theory for nonsingular injective modules provided by Goodearl and Boyle.
Ebrahim Hashemi - One of the best experts on this subject based on the ideXlab platform.
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Baer AND QUASI-Baer PROPERTIES OF SKEW PBW EXTENSIONS
2019Co-Authors: Ebrahim Hashemi, Kh. Khalilnezhad, M. GhadiriAbstract:A ring $R$ with an automorphism $sigma$ and a $sigma$-derivation $delta$ is called $delta$-quasi-Baer (resp., $sigma$-invariant quasi-Baer) if the right annihilator of every $delta$-ideal (resp., $sigma$-invariant ideal) of $R$ is generated by an idempotent, as a right ideal. In this paper, we study Baer and quasi-Baer properties of skew PBW extensions. More exactly, let $A=sigma(R)leftlangle x_{1},ldots,x_{n}rightrangle $ be a skew PBW extension of derivation type of a ring $R$. (i) It is shown that $ R$ is $Delta$-quasi-Baer if and only if $ A$ is quasi-Baer.(ii) $ R$ is $Delta$-Baer if and only if $ A$ is Baer, when $R$ has IFP. Also, let $A=sigma (R)leftlangle x_1, ldots , x_nrightrangle$ be a quasi-commutative skew PBW extension of a ring $R$. (iii) If $R$ is a $Sigma$-quasi-Baer ring, then $A $ is a quasi-Baer ring. (iv) If $A $ is a quasi-Baer ring, then $R$ is a $Sigma$-invariant quasi-Baer ring. (v) If $R$ is a $Sigma$-Baer ring, then $A $ is a Baer ring, when $R$ has IFP. (vi) If $A $ is a Baer ring, then $R$ is a $Sigma$-invariant Baer ring. Finally, we show that if $A = sigma (R)leftlangle x_1, ldots , x_nrightrangle $ is a bijective skew PBW extension of a quasi-Baer ring $R$, then $A$ is a quasi-Baer ring.
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On extensions of Baer and quasi-Baer modules
Acta Universitatis Sapientiae Mathematica, 2018Co-Authors: Ebrahim Hashemi, Marzieh Yazdanfar, Abdollah AlhevazAbstract:Abstract Let R be a ring, MR a module, S a monoid, ω : S → End(R) a monoid homomorphism and R * S a skew monoid ring. Then M[S] = {m1g1 + · · · + mngn | n ≥ 1, mi ∈ M and gi ∈ S for each 1 ≤ i ≤ n} is a module over R ∗ S. A module MR is Baer (resp. quasi-Baer) if the annihilator of every subset (resp. submodule) of M is generated by an idempotent of R. In this paper we impose S-compatibility assumption on the module MR and prove: (1) MR is quasi-Baer if and only if M[s]R∗S is quasi-Baer, (2) MR is Baer (resp. p.p) if and only if M[S]R∗S is Baer (resp. p.p), where MR is S-skew Armendariz, (3) MR satisfies the ascending chain condition on annihilator of submodules if and only if so does M[S]R∗S, where MR is S-skew quasi-Armendariz.
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Extensions of Baer and quasi-Baer modules
Bulletin of The Iranian Mathematical Society, 2011Co-Authors: Ebrahim HashemiAbstract:We study the relationships between the Baer, quasi- Baer and p.q.-Baer property of an R-module M and the polynomial extensions of module M. As a consequence of our results, we obtain some results of (C.Y. Hong, N.K. Kim and T.K. Kwak, J. Pure Appl. Algebra 151 (2000) 215-226.) and (E. Hashemi and A. Moussavi, Acta Math. Hungar. 107 (2005) 207-224.).
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Generalized Quasi-Baer Rings
Communications in Algebra, 2005Co-Authors: Ahmad Moussavi, Hamid Haj Seyyed Javadi, Ebrahim HashemiAbstract:ABSTRACT We say a ring with identity is a generalized right (principally) quasi-Baer if for any (principal) right ideal I of R, the right annihilator of In is generated by an idempotent for some positive integer n, depending on I. The behavior of the generalized right (principally) quasi-Baer condition is investigated with respect to various constructions and extensions. The class of generalized right (principally) quasi-Baer rings includes the right (principally) quasi-Baer rings and is closed under direct product and also under some kinds of upper triangular matrix rings. The generalized right (principally) quasi-Baer condition is a Morita invariant property. Examples to illustrate and delimit the theory are provided.
