• Title, Summary, Keyword: quasi-injective

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SOME NEW CHARACTERIZATIONS OF QUASI-FROBENIUS RINGS BY USING PURE-INJECTIVITY

  • Moradzadeh-Dehkordi, Ali
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.2
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    • pp.371-381
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    • 2020
  • A ring R is called right pure-injective if it is injective with respect to pure exact sequences. According to a well known result of L. Melkersson, every commutative Artinian ring is pure-injective, but the converse is not true, even if R is a commutative Noetherian local ring. In this paper, a series of conditions under which right pure-injective rings are either right Artinian rings or quasi-Frobenius rings are given. Also, some of our results extend previously known results for quasi-Frobenius rings.

THE JACOBSON RADICAL OF THE ENDOMORPHISM RING, THE JACOBSON RADICAL, AND THE SOCLE OF AN ENDO-FLAT MODULE

  • Bae, Soon-Sook
    • Communications of the Korean Mathematical Society
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    • v.15 no.3
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    • pp.453-467
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    • 2000
  • For any S-flat module RM(which will be called endoflat) with a commutaitve ring R with identity, where S is the endomorphism ring RM, the fact that every epimorphism is an automorphism has been proved and the Jacobson Radical Rad(S) of S is described as follow; Rad(S) = { f$\in$S|Imf=Mf is small in M} = {f$\in$S|Imf $\leq$Rad(M)}. Additionally for any quasi-injective endo-flat module RM, the fact that every monomorphism is an automorphism has been proved and the Jacobson Radical Rad(S) for any quasi-injective endo-flat module has been studied too. Also some equivalent conditions for the semi-primitivity of any faithful endo-flat module RM with the open Jacobson Radical Rad(M) and those for the semi-simplicity of any faithful endo-flat quasi-injective module RM with the closed Socle Soc(M) have been studied.

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CERTAIN DISCRIMINATIONS OF PRIME ENDOMORPHISM AND PRIME MATRIX

  • Bae, Soon-Sook
    • East Asian mathematical journal
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    • v.14 no.2
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    • pp.259-268
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    • 1998
  • In this paper, for a commutative ring R with an identity, considering the endomorphism ring $End_R$(M) of left R-module $_RM$ which is (quasi-)injective or (quasi-)projective, some discriminations of prime endomorphism were found as follows: each epimorphism with the irreducible(or simple) kernel on a (quasi-)injective module and each monomorphism with maximal image on a (quasi-)projective module are prime. It was shown that for a field F, any given square matrix in $Mat_{n{\times}n}$(F) with maximal image and irreducible kernel is a prime matrix, furthermore, any given matrix in $Mat_{n{\times}n}$(F) for any field F can be factored into a product of prime matrices.

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GENERALIZATIONS OF THE QUASI-INJECTIVE MODULE

  • Han, Chang-Woo;Park, Su-Jeong
    • Communications of the Korean Mathematical Society
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    • v.10 no.4
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    • pp.811-813
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    • 1995
  • The purpose of this paper is to prove the divisibility of a direct injective module and every closed submodule of a $\pi$-injective module M is a direct summand of M.

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QUASI SIMILARITY AND INJECTIVE p-QUASIHYPONORMAL OPERATORS

  • Woo, Young-Jin
    • Bulletin of the Korean Mathematical Society
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    • v.42 no.3
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    • pp.653-659
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    • 2005
  • In this paper it is proved that quasisimilar n-tuples of tensor products of injective p-quasihyponormal operators have the same spectra, essential spectra and indices, respectively. And it is also proved that a Weyl n-tuple of tensor products of injective p-quasihyponormal operators can be perturbed by an n-tuple of compact operators to an invertible n-tuple.

A STUDY ON GENERALIZED QUASI-CLASS A OPERATORS

  • Kim, Geon-Ho;Jeon, In-Ho
    • Korean Journal of Mathematics
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    • v.17 no.2
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    • pp.155-159
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    • 2009
  • In this paper, we consider the operator T satisfying $T^{*k}({\mid}T^2{\mid}-{\mid}T{\mid}^2)T^k{\geq}0$ and prove that if the operator is injective and has the real spectrum, then it is self-adjoint.

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ON INJECTIVITY AND P-INJECTIVITY, IV

  • Chi Ming, Roger Yue
    • Bulletin of the Korean Mathematical Society
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    • v.40 no.2
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    • pp.223-234
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    • 2003
  • This note contains the following results for a ring A : (1) A is simple Artinian if and only if A is a prime right YJ-injective, right and left V-ring with a maximal right annihilator ; (2) if A is a left quasi-duo ring with Jacobson radical J such that $_{A}$A/J is p-injective, then the ring A/J is strongly regular ; (3) A is von Neumann regular with non-zero socle if and only if A is a left p.p.ring containing a finitely generated p-injective maximal left ideal satisfying the following condition : if e is an idempotent in A, then eA is a minimal right ideal if and only if Ae is a minimal left ideal ; (4) If A is left non-singular, left YJ-injective such that each maximal left ideal of A is either injective or a two-sided ideal of A, then A is either left self-injective regular or strongly regular : (5) A is left continuous regular if and only if A is right p-injective such that for every cyclic left A-module M, $_{A}$M/Z(M) is projective. ((5) remains valid if 《continuous》 is replaced by 《self-injective》 and 《cyclic》 is replaced by 《finitely generated》. Finally, we have the following two equivalent properties for A to be von Neumann regula. : (a) A is left non-singular such that every finitely generated left ideal is the left annihilator of an element of A and every principal right ideal of A is the right annihilator of an element of A ; (b) Change 《left non-singular》 into 《right non-singular》in (a).(a).