• The Pure and Applied Mathematics
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• v.10 no.4
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• pp.199-206
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• 2003

Let N be a right near-ring. N is said to be strongly reduced if, for $a\inN$, $a^2 \in N_{c}$ implies $a\;\in\;N_{c}$, or equivalently, for $a\inN$ and any positive integer n, $a^{n} \in N_{c}$ implies $a\;\in\;N_{c}$, where $N_{c}$ denotes the constant part of N. We will show that strong reducedness is equivalent to condition (ⅱ) of Reddy and Murty's property $(^{\ast})$ (cf. [Reddy & Murty: On strongly regular near-rings. Proc. Edinburgh Math. Soc. (2) 27 (1984), no. 1, 61-64]), and that condition (ⅰ) of Reddy and Murty's property $(^{\ast})$ follows from strong reducedness. Also, we will investigate some characterizations of strongly reduced near-rings and their properties. Using strong reducedness, we characterize left strongly regular near-rings and ($P_{0}$)-near-rings.

• Journal of the Korean Mathematical Society
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• v.48 no.6
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• pp.1189-1201
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• 2011

Let R be a ring and M a right R-module, S = $End_R$(M). The module M is called almost principally small injective (or APS-injective for short) if, for any a ${\in}$ J(R), there exists an S-submodule $X_a$ of M such that $l_Mr_R$(a) = Ma $Ma{\bigoplus}X_a$ as left S-modules. If $R_R$ is a APS-injective module, then we call R a right APS-injective ring. We develop, in this paper, APS-injective rings as a generalization of PS-injective rings and AP-injective rings. Many examples of APS-injective rings are listed. We also extend some results on PS-injective rings and AP-injective rings to APS-injective rings.

• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.945-956
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• 2020

We introduce the concept of S-exchange rings to unify various subclass of exchange rings, where S is a subset of the ring. Many properties on S-exchange rings are obtained. For instance, we show that a ring R is clean if and only if R is left U(R)-exchange, a ring R is nil clean if and only if R is left (N(R) - 1)-exchange, and that a ring R is J-clean if and only if R is left (J(R) - 1)-exchange. As a conclusion, we obtain a sufficient condition such that clean (nil clean property, respectively) can pass to corners and reprove that J-clean passes to corners by a different way.

• Journal of the Korean Mathematical Society
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• v.47 no.5
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• pp.879-897
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• 2010

Y. Hirano introduced the concept of a quasi-Armendariz ring which extends both Armendariz rings and semiprime rings. A ring R is called quasi-Armendariz if $a_iRb_j$ = 0 for each i, j whenever polynomials $f(x)\;=\;\sum_{i=0}^ma_ix^i$, $g(x)\;=\;\sum_{j=0}^mb_jx^j\;{\in}\;R[x]$ satisfy f(x)R[x]g(x) = 0. In this paper, we first extend the quasi-Armendariz property of semiprime rings to the skew polynomial rings, that is, we show that if R is a semiprime ring with an epimorphism $\sigma$, then f(x)R[x; $\sigma$]g(x) = 0 implies $a_iR{\sigma}^{i+k}(b_j)=0$ for any integer k $\geq$ 0 and i, j, where $f(x)\;=\;\sum_{i=0}^ma_ix^i$, $g(x)\;=\;\sum_{j=0}^mb_jx^j\;{\in}\;R[x,\;{\sigma}]$. Moreover, we extend this property to the skew monoid rings, the Ore extensions of several types, and skew power series ring, etc. Next we define $\sigma$-skew quasi-Armendariz rings for an endomorphism $\sigma$ of a ring R. Then we study several extensions of $\sigma$-skew quasi-Armendariz rings which extend known results for quasi-Armendariz rings and $\sigma$-skew Armendariz rings.

• Journal of the Chungcheong Mathematical Society
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• v.29 no.4
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• pp.643-650
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• 2016

In this paper, we introduce the notion of a semiderivation on near-rings, and we try to generalize some properties of prime rings with derivations to prime near-rings with semiderivations.

• Korean Journal of Mathematics
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• v.9 no.1
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• pp.1-7
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• 2001

In this paper, we define and study Gauss sums over Galois rings of characteristic 4. In particular, we give the absolute value of Gauss sum over Galois rings of characteristic 4.

• East Asian mathematical journal
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• v.24 no.2
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• pp.145-149
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• 2008

From the notion of bi-ideals in near-rings, various generalizations of regularity conditions have been studied. In this paper, we generalize further the notion of bi-ideals and introduce the notion of weak bi-ideals in near-rings and obtain some characterizations using this concept in left self distributive near-rings.

• Journal of the Chungcheong Mathematical Society
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• v.14 no.2
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• pp.19-26
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• 2001

All rings are assumed to be associative but do not necessarily have an identity. In this paper, we carry out a study of ring theoretic properties of formal triangular matrix rings. Some results are obtained on these rings concerning properties such as being $I_0$-ring, I-ring, exchange ring.

• Communications of the Korean Mathematical Society
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• v.39 no.2
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• pp.267-278
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• 2024

In this paper, we continue to explore an idea presented in [3] and introduce a new class of matrix rings called staircase matrix rings which has applications in noncommutative ring theory. We show that these rings preserve the notions of reduced, symmetric, reversible, IFP, reflexive, abelian rings, etc.

• Bulletin of the Korean Mathematical Society
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• v.45 no.2
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• pp.285-297
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• 2008

For an endomorphism ${\alpha}$ of a ring R, the endomorphism ${\alpha}$ is called semicommutative if ab=0 implies $aR{\alpha}(b)$=0 for a ${\in}$ R. A ring R is called ${\alpha}$-semicommutative if there exists a semicommutative endomorphism ${\alpha}$ of R. In this paper, various results of semicommutative rings are extended to ${\alpha}$-semicommutative rings. In addition, we introduce the notion of an ${\alpha}$-skew power series Armendariz ring which is an extension of Armendariz property in a ring R by considering the polynomials in the skew power series ring $R[[x;\;{\alpha}]]$. We show that a number of interesting properties of a ring R transfer to its the skew power series ring $R[[x;\;{\alpha}]]$ and vice-versa such as the Baer property and the p.p.-property, when R is ${\alpha}$-skew power series Armendariz. Several known results relating to ${\alpha}$-rigid rings can be obtained as corollaries of our results.