• Honam Mathematical Journal
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• v.39 no.3
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• pp.317-330
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• 2017

In this paper we introduce some new kinds of injectivities, namely, LC (resp. Ind, PInd) injectivity and investigate the relation among various kinds of injectivities. Some classifications of monoids by properties of these kinds of injective acts are presented. Among other results, it is shown that over a principal right ideal monoid, right completely LC-injectivity implies right completely injectivity. Also over a monoid with a zero Ind-injective (resp. PInd-injective) acts are injective.

• Korean Journal of Mathematics
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• v.25 no.4
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• pp.579-585
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• 2017

Based on injectivity, we introduce some definitions in the category NFHG of normal fuzzy hypergroups. And we show that a complete normal fuzzy hypergroup is a weakly injective object in NFHG. Also we investigate weak injectivity in the comma category NFHG/K.

• Kyungpook Mathematical Journal
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• v.51 no.2
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• pp.177-185
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• 2011

A right ideal I of a ring R is small in case for every proper right ideal K of R, K + I ${\neq}$ = R. A right R-module M is called PS-injective if every R-homomorphism f : aR ${\rightarrow}$ M for every principally small right ideal aR can be extended to R ${\rightarrow}$ M. A ring R is called right PS-injective if R is PS-injective as a right R-module. We develop, in this article, PS-injectivity as a generalization of P-injectivity and small injectivity. Many characterizations of right PS-injective rings are studied. In light of these facts, we get several new properties of a right GPF ring and a semiprimitive ring in terms of right PS-injectivity. Related examples are given as well.

• Communications of the Korean Mathematical Society
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• v.23 no.4
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• pp.541-548
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• 2008

For a completely bounded linear maps between operator spaces, we introduce numbers which measure the degree of injectivity and subjectivity. The number measuring the injectivity is an operator space analogue of the minimum modulus of a linear map in normed spaces.

• Bulletin of the Korean Mathematical Society
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• v.39 no.2
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• pp.251-258
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• 2002

Let R be a commutative Noetherian ring and M an R-module. In this paper we will consider the asymptotic behaviour of ideals relative to an R-module E which is M-injective.

• Journal of the Korean Mathematical Society
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• v.52 no.1
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• pp.67-80
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• 2015

For a pomonoid S, let us denote Pos-S the category of S-posets and S-poset maps. In this paper, we consider the slice category Pos-S/B for an S-poset B, and study some categorical ingredients. We first show that there is no non-trivial injective object in Pos-S/B. Then we investigate injective objects with respect to the class of regular monomorphisms in this category and show that Pos-S/B has enough regular injective objects. We also prove that regular injective objects are retracts of exponentiable objects in this category. One of the main aims of the paper is to draw attention to characterizing injectivity in the category Pos-S/B under a particular case where B has trivial action. Among other things, we also prove that the necessary condition for a map (an object) here to be regular injective is being convex and present an example to show that the converse is not true, in general.

• Kyungpook Mathematical Journal
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• v.62 no.4
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• pp.657-671
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• 2022

Where S is a pomonoid, let Pos-S be the category of S-posets and S-poset maps. We start off by characterizing the pomonoids S for which all projectives in this category are either generators or free. We then study the notions of regular injectivity and weakly regularly d-injectivity in this category. This leads to homological classification results for pomonoids. Among other things, we get find relationships between regular injectivity in the slice category Pos-S/BS, for any S-poset BS, and generators and cyclic projectives in Pos-S.

• Kyungpook Mathematical Journal
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• v.47 no.2
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• pp.267-276
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• 2007

Injective JW-algebras are defined and are characterized by the existence of projections of norm 1 onto them. The relationship between the injectivity of a JW-algebra and the injectivity of its universal enveloping von Neumann algebra is established. The Jordan analgue of Theorem 3 of [3] is proved, that is, a JC-algebra A is nuclear if and only if its second dual $A^{**}$ is injective.

• Bulletin of the Korean Mathematical Society
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• v.61 no.2
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• pp.301-315
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• 2024

Some results are generalized from principally injective rings to principally injective modules. Moreover, it is proved that the results are valid to some other extended injectivity conditions which may be defined over modules. The influence of such injectivity conditions are studied for both the trace and the reject submodules of some modules over commutative rings. Finally, a correction is given to a paper related to the subject.

• 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).