Properties inherited by direct sums of copies of a module
-
Jae Keol Park
Abstract
It is known that the quasi-retractable property of modules helps characterize a Baer module in terms of the Baer property of its endomorphism ring. We first study conditions which allow us to obtain the quasi-retractable property for certain direct sums of copies of a quasi-retractable module. Then if a ring R is right nonsingular for which the right and the left maximal rings of quotients coincide (e.g., R is semiprime PI) and M is an intermediate (R, R)-bimodule between R and the maximal right ring of quotients Q(R) of R, it is shown that for any given positive integer n, M(n) R is Kcononsingular. As an application, we prove that M(n) R is a Baer module if and only if , M(n) R is an extending module for any positive integer n. In particular, if R is a right nonsingular ring for which the right and the left maximal rings of quotients coincide and A is a right ring of quotients of R, then A(n)R is a Baer module if and only if A(n)R is an extending module for any positive integer n. Examples which illustrate and delimit our results are provided
Abstract
It is known that the quasi-retractable property of modules helps characterize a Baer module in terms of the Baer property of its endomorphism ring. We first study conditions which allow us to obtain the quasi-retractable property for certain direct sums of copies of a quasi-retractable module. Then if a ring R is right nonsingular for which the right and the left maximal rings of quotients coincide (e.g., R is semiprime PI) and M is an intermediate (R, R)-bimodule between R and the maximal right ring of quotients Q(R) of R, it is shown that for any given positive integer n, M(n) R is Kcononsingular. As an application, we prove that M(n) R is a Baer module if and only if , M(n) R is an extending module for any positive integer n. In particular, if R is a right nonsingular ring for which the right and the left maximal rings of quotients coincide and A is a right ring of quotients of R, then A(n)R is a Baer module if and only if A(n)R is an extending module for any positive integer n. Examples which illustrate and delimit our results are provided
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents VII
- On structure of ∗-prime rings with generalized derivation 1
- A characterization of additive mappings in rings with involution 11
- Skew constacyclic codes over Fq + vFq + v2Fq 25
- Generalized total graphs of commutative rings: a survey 37
- Differential conditions for which near-rings are commutative rings 55
- Generalized skew derivations satisfying the second posner’s theorem on lie ideals 65
- Generalized skew-derivations on lie ideals in prime rings 81
- On generalized derivations and commutativity of prime rings with involution 91
- On (n, d)-krull property in amalgamated algebra 101
- Pure ideals in ordered Γ-semigroups 111
- Projective ideals of differential polynomial rings over hnp rings 121
- Additive central m-power skew-commuting maps on semiprime rings 135
- A note on cess-lattices 151
- Properties inherited by direct sums of copies of a module 163
- Modules witnessing that a leavitt path algebra is directly infinite 181
- Inductive groupoids and normal categories of regular semigroups 193
- Actions of generalized derivations in rings and banach algebras 201
- Proper categories and their duals 215
- On nakayama conjecture and related conjectures-review 227
- On construction of global actions for partial actions 243
- On 2-absorbing and weakly 2-absorbing ideals in product lattices 253
- Separability in algebra and category theory 265
- Annihilators of power values of generalized skew derivations on lie ideals 307
- Generalized derivations on prime rings with involution 317
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents VII
- On structure of ∗-prime rings with generalized derivation 1
- A characterization of additive mappings in rings with involution 11
- Skew constacyclic codes over Fq + vFq + v2Fq 25
- Generalized total graphs of commutative rings: a survey 37
- Differential conditions for which near-rings are commutative rings 55
- Generalized skew derivations satisfying the second posner’s theorem on lie ideals 65
- Generalized skew-derivations on lie ideals in prime rings 81
- On generalized derivations and commutativity of prime rings with involution 91
- On (n, d)-krull property in amalgamated algebra 101
- Pure ideals in ordered Γ-semigroups 111
- Projective ideals of differential polynomial rings over hnp rings 121
- Additive central m-power skew-commuting maps on semiprime rings 135
- A note on cess-lattices 151
- Properties inherited by direct sums of copies of a module 163
- Modules witnessing that a leavitt path algebra is directly infinite 181
- Inductive groupoids and normal categories of regular semigroups 193
- Actions of generalized derivations in rings and banach algebras 201
- Proper categories and their duals 215
- On nakayama conjecture and related conjectures-review 227
- On construction of global actions for partial actions 243
- On 2-absorbing and weakly 2-absorbing ideals in product lattices 253
- Separability in algebra and category theory 265
- Annihilators of power values of generalized skew derivations on lie ideals 307
- Generalized derivations on prime rings with involution 317
