• Communications of the Korean Mathematical Society
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• v.30 no.4
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• pp.363-377
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• 2015

Let R be a ring, (S,${\leq}$) a strictly ordered monoid and ${\omega}$ : S ${\rightarrow}$ End(R) a monoid homomorphism. The skew generalized power series ring R[[S,${\omega}$]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev-Neumann Laurent series rings. In this paper, we investigate the interplay between the ring-theoretical properties of R[[S,${\omega}$]] and the graph-theoretical properties of its zero-divisor graph ${\Gamma}$(R[[S,${\omega}$]]). Furthermore, we examine the preservation of diameter and girth of the zero-divisor graph under extension to skew generalized power series rings.

• Bulletin of the Korean Mathematical Society
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• v.30 no.1
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• pp.71-77
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• 1993

Every ring considered in the paper will be assumed to be commutative and have a unit element. An ideal A of a ring R will be called primal if the elements of R which are zero divisors modulo A, form an ideal of R, say pp. If A is a primal ideal of R, P is called the adjoint ideal of A. The adjoint ideal of a primal ideal is prime [2]. The definition of primal ideals may also be formulated as follows: An ideal A of a ring R is primal if in the residue class ring R/A the zero divisors form an ideal of R/A. If Q is a primary idel of a ring R then every zero divisor of R/Q is nilpotent; therefore, Q is a primal ideal of R. That a primal ideal need not be primary, is shown by an example in [2]. Let R[X], and R[[X]] denote the polynomial ring and formal power series ring in an indeterminate X over a ring R, respectively. Let S be a multiplicative system in a ring R and S$^{-1}$ R the quotient ring of R. Let Q be a P-primary ideal of a ring R. Then Q[X] is a P[X]-primary ideal of R[X], and S$^{-1}$ Q is a S$^{-1}$ P-primary ideal of a ring S$^{-1}$ R if S.cap.P=.phi., and Q[[X]] is a P[[X]]-primary ideal of R[[X]] if R is Noetherian [1]. We search for analogous results when primary ideals are replaced with primal ideals. To show an ideal A of a ring R to be primal, it sufficies to show that a-b is a zero divisor modulo A whenever a and b are zero divisors modulo A.

• Kyungpook Mathematical Journal
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• v.53 no.3
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• pp.397-406
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• 2013

A ring R is called a left (right) SF-ring if simple left (right) R-modules are flat. It is still unknown whether a left (right) SF-ring is von Neumann regular. In this paper, we give some conditions for a left (right) SF-ring to be (a) von Neumann regular; (b) strongly regular; (c) division ring. It is proved that: (1) a right SF-ring R is regular if maximal essential right (left) ideals of R are weakly left (right) ideals of R (this result gives an affirmative answer to the question raised by Zhang in 1994); (2) a left SF-ring R is strongly regular if every non-zero left (right) ideal of R contains a non-zero left (right) ideal of R which is a W-ideal; (3) if R is a left SF-ring such that $l(x)(r(x))$ is an essential left (right) ideal for every right (left) zero divisor x of R, then R is a division ring.

• Bulletin of the Korean Mathematical Society
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• v.59 no.5
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• pp.1237-1246
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• 2022

In this paper, we introduce the concept of N-pure ideal as a generalization of pure ideal. Using this concept, a new and interesting type of rings is presented, we call it a mid ring. Also, we provide new characterizations for von Neumann regular and zero-dimensional rings. Moreover, some results about mp-ring are given. Finally, a characterization for mid rings is provided. Then it is shown that the class of mid rings is strictly between the class of reduced mp-rings (p.f. rings) and the class of mp-rings.

• Bulletin of the Korean Mathematical Society
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• v.61 no.3
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• pp.763-777
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• 2024

Let R be a commutative ring with identity 1≠ 0. In this paper, we continue the study started in [10] to further investigate when the extended zero-divisor graph of R, denoted as $\bar{\Gamma}$(R), is complemented. We also study when $\bar{\Gamma}$(R) is uniquely complemented. We give a complete characterization of when $\bar{\Gamma}$(R) of a finite ring R is complemented. Various examples are given using the direct product of rings and idealizations of modules.

• Journal of the Korean Mathematical Society
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• v.53 no.2
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• pp.381-401
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• 2016

According to Jacobson [31], a right ideal is bounded if it contains a non-zero ideal, and Faith [15] called a ring strongly right bounded if every non-zero right ideal is bounded. From [30], a ring is strongly right AB if every non-zero right annihilator is bounded. In this paper, we introduce and investigate a particular class of McCoy rings which satisfy Property (A) and the conditions asked by Nielsen [42]. It is shown that for a u.p.-monoid M and ${\sigma}:M{\rightarrow}End(R)$ a compatible monoid homomorphism, if R is reversible, then the skew monoid ring R * M is strongly right AB. If R is a strongly right AB ring, M is a u.p.-monoid and ${\sigma}:M{\rightarrow}End(R)$ is a weakly rigid monoid homomorphism, then the skew monoid ring R * M has right Property (A).

• Korean Journal of Mathematics
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• v.29 no.1
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• pp.13-24
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• 2021

For a finite commutative ring R with identity 1 ≠ 0, the zero divisor graph ��(R) is a simple connected graph having vertex set as the set of nonzero zero divisors of R, where two vertices x and y are adjacent if and only if xy = 0. We find the signless Laplacian spectrum of the zero divisor graphs ��(ℤn) for various values of n. Also, we find signless Laplacian spectrum of ��(ℤn) for n = pz, z ≥ 2, in terms of signless Laplacian spectrum of its components and zeros of the characteristic polynomial of an auxiliary matrix. Further, we characterise n for which zero divisor graph ��(ℤn) are signless Laplacian integral.

• Bulletin of the Korean Mathematical Society
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• v.59 no.6
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• pp.1557-1565
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• 2022

In this paper, Glift codes, generalized lifted polynomials, matrices are introduced. The advantage of Glift code is "distance preserving" over the ring R. Then optimal codes can be obtained over the rings by using Glift codes and lifted polynomials. Zero divisors are classified to satisfy "distance preserving" for codes over non-chain rings. Moreover, Glift codes apply on MDS codes and MDS codes are obtained over the ring 𝓡 and the non-chain ring 𝓡_e,s.

• Journal of the Korean Mathematical Society
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• v.53 no.3
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• pp.519-532
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• 2016

The zero-divisor graph of a noncommutative ring R, denoted by ${\Gamma}(R)$, is a graph whose vertices are nonzero zero-divisors of R, and there is a directed edge from a vertex x to a distinct vertex y if and only if xy = 0. Let $R=M_2(F_q)$ be the $2{\times}2$ matrix ring over a finite field $F_q$. In this article, we investigate the automorphism group of ${\Gamma}(R)$.

• Kyungpook Mathematical Journal
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• v.62 no.3
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• pp.437-453
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• 2022

Unlike for polynomial rings, the notion of multiplication for the near-ring of polynomials is the substitution operation. This leads to somewhat surprising results. Let S be an abelian left near-ring with identity. The relation ~ on S defined by letting a ~ b if and only if ann_S(a) = ann_S(b), is an equivalence relation. The compressed zero-divisor graph 𝚪_E(S) of S is the undirected graph whose vertices are the equivalence classes induced by ~ on S other than [0]_S and [1]_S, in which two distinct vertices [a]_S and [b]_S are adjacent if and only if ab = 0 or ba = 0. In this paper, we are interested in studying the compressed zero-divisor graphs of the zero-symmetric near-ring of polynomials R₀[x] and the near-ring of the power series R₀[[x]] over a commutative ring R. Also, we give a complete characterization of the diameter of these two graphs. It is natural to try to find the relationship between diam(𝚪_E(R₀[x])) and diam(𝚪_E(R₀[[x]])). As a corollary, it is shown that for a reduced ring R, diam(𝚪_E(R)) ≤ diam(𝚪_E(R₀[x])) ≤ diam(𝚪_E(R₀[[x]])).