• Korean Journal of Mathematics
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• v.10 no.2
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• pp.117-122
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• 2002

In this paper, we give some properties of left(right) semi-regular and $g$-regular $po$-semigroups.

• 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).

• Kyungpook Mathematical Journal
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• v.62 no.1
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• pp.29-41
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• 2022

We study the concept of a quasi-ideal and a generalized bi-ideal of an n-ary semigroup; give a construction of the quasi-ideal of an n-ary semigroup generated by its nonempty subset; and introduce the notions of regularities, namely, a left regularity and a left weakly regularity. Moreover, the notions of a right regularity, a right weak regularity and a complete regularity are given. Finally, characterizations of these regularities are presented.

• Honam Mathematical Journal
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• v.36 no.4
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• pp.885-893
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• 2014

In this paper, we introduce the concept of a right f-derivation in incline algebras and give some properties of incline algebras. Also, the concept of d-ideal is introduced in an incline algebra with respect to right f-derivations.

• Journal of the Chungcheong Mathematical Society
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• v.34 no.2
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• pp.119-129
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• 2021

In this paper, we introduce the concept of a generalized right f-derivation associated with a derivation d and a function f in 𝚪-incline algebras and give some properties of 𝚪-incline algebras. Also, the concept of d-ideal is introduced in a 𝚪-incline algebra with respect to right f-derivations.

• Honam Mathematical Journal
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• v.28 no.4
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• pp.485-497
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• 2006

Further properties on a fuzzy ideal of a right K(G)-algbera $\mathcal{G}$ are investigated. Using a family of ideals of a right K(G)-algebra $\mathcal{G}$ with additional conditions, a fuzzy ideal of $\mathcal{G}$ is established. Given a fuzzy set $\mu$ in $\mathcal{G}$, the least fuzzy ideal of $\mathcal{G}$ containing $\mu$ is described. Using a chain of ideals of $\mathcal{G}$, a fuzzy ideal of $\mathcal{G}$ is constructed, and their properties are investigated.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.43 no.2
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• pp.299-307
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• 2006

The following results ale extended from P-injective rings to AP-injective rings: (1) R is left self-injective regular if and only if R is a right (resp. left) AP-injective ring such that for every finitely generated left R-module M, $_R(M/Z(M))$ is projective, where Z(M) is the left singular submodule of $_{R}M$; (2) if R is a left nonsingular left AP-injective ring such that every maximal left ideal of R is either injective or a two-sided ideal of R, then R is either left self-injective regular or strongly regular. In addition, we answer a question of Roger Yue Chi Ming [13] in the positive. Let R be a ring whose every simple singular left R-module is Y J-injective. If R is a right MI-ring whose every essential right ideal is an essential left ideal, then R is a left and right self-injective regular, left and right V-ring of bounded index.

• Kyungpook Mathematical Journal
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• v.47 no.4
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• pp.473-480
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• 2007

In this paper we define prime right ideals of semirings and prove that if every right ideal of a semiring R is prime then R is weakly regular. We also prove that if the set of right ideals of R is totally ordered then every right ideal of R is prime if and only if R is right weakly regular. Moreover in this paper we also define prime subsemimodule (generalizing the concept of prime right ideals) of an R-semimodule. We prove that if a subsemimodule K of an R-semimodule M is prime then $A_K(M)$ is also a prime ideal of R.

• East Asian mathematical journal
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• v.6 no.2
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• pp.167-182
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• 1990

The relationships between submodules of a module and ideals of the endomorphism ring of a module had been studied in [1]. For a submodule L of a moudle M, the set $I^L$ of all endomorphisms whose images are contained in L is a left ideal of the endomorphism ring End (M) and for a submodule N of M, the set $I_N$ of all endomorphisms whose kernels contain N is a right ideal of End (M). In this paper, author defines an H-invariant module and proves that every submodule of an H-invariant module is the image and kernel of unique endomorphisms. Every ideal $I^L(I_N)$ of the endomorphism ring End(M) when M is H-invariant is a left (respectively, right) principal ideal of End(M). From the above results, if a module M is H-invariant then each left, right, or both sided ideal I of End(M) is an intersection of a left, right, or both sided principal ideal and I itself appropriately. If M is an H-invariant module then the ACC on the set of all left ideals of type $I^L$ implies the ACC on M. Also if the set of all right ideals of type $I^L$ has DCC, then H-invariant module M satisfies ACC. If the set of all left ideals of type $I^L$ satisfies DCC, then H-invariant module M satisfies DCC. If the set of all right ideals of type $I_N$ satisfies ACC then H-invariant module M satisfies DCC. Therefore for an H-invariant module M, if the endomorphism ring End(M) is left Noetherian, then M satisfies ACC. And if End(M) is right Noetherian then M satisfies DCC. For an H-invariant module M, if End(M) is left Artinian then M satisfies DCC. Also if End(M) is right Artinian then M satisfies ACC.