• 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 ${\Gamma}(R)$.

• Journal of the Chungcheong Mathematical Society
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• v.30 no.1
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• pp.77-102
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• 2017

As an analogy of $Poincar{\acute{e}}$ series in the space of modular forms, T. Ono associated a module $M_c/P_c$ for ${\gamma}=[c]{\in}H^1(G,R^{\times})$ where finite group G is acting on a ring R. $M_c/P_c$ is regarded as the 0-dimensional twisted Tate cohomology ${\hat{H}}^0(G,R^+)_{\gamma}$. In the case that G is the Galois group of a Galois extension K of a number field k and R is the ring of integers of K, the vanishing properties of $M_c/P_c$ are related to the ramification of K/k. We generalize this to arbitrary n-dimensional twisted cohomology of the ring of integers and obtain the extended version of theorems. Moreover, some explicit examples on quadratic and biquadratic number fields are given.

• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.131-150
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• 2019

In this paper, we consider the structure of ${\gamma}$-constacyclic codes of length $2p^s$ over the Galois ring $GR(p^a,m)$ for any unit ${\gamma}$ of the form ${\xi}_0+p{\xi}_1+p^2z$, where $z{\in}GR(p^a,m)$ and ${\xi}_0$, ${\xi}_1$ are nonzero elements of the set ${\mathcal{T}}(p,m)$. Here ${\mathcal{T}}(p,m)$ denotes a complete set of representatives of the cosets ${\frac{GR(p^a,m)}{pGR(p^a,m)}}={\mathbb{F}}p^m$ in $GR(p^a,m)$. When ${\gamma}$ is not a square, the rings ${\mathcal{R}}_p(a,m,{\gamma})=\frac{GR(p^a,m)[x]}{{\langle}x^2p^s-{\gamma}{\rangle}}$ is a chain ring with maximal ideal ${\langle}x^2-{\delta}{\rangle}$, where ${\delta}p^s={\xi}_0$, and the number of codewords of ${\gamma}$-constacyclic code are provided. Furthermore, the self-orthogonal and self-dual ${\gamma}$-constacyclic codes of length $2p^s$ over $GR(p^a,m)$ are also established. Finally, we determine the Rosenbloom-Tsfasman (RT) distances and weight distributions of all such codes.

• The Pure and Applied Mathematics
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• v.13 no.2 s.32
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• pp.113-120
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• 2006

In this paper, we introduce the concept of P(r,m) $\Gamma$-near-ring and obtain some characterization of P(r,m) $\Gamma$-near-rings through regularity conditions.

• Journal of applied mathematics & informatics
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• v.18 no.1_2
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• pp.45-57
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• 2005

An efficient algorithm for finding the inverse and the group inverse of the FLS $\gamma-circulant$ matrix is presented by Euclidean algorithm. Extension is made to compute the inverse of the FLS $\gamma-retrocirculant$ matrix by using the relationship between an FLS $\gamma-circulant$ matrix and an FLS $\gamma-retrocirculant$ matrix. Finally, some examples are given.

• Kyungpook Mathematical Journal
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• v.50 no.3
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• pp.411-417
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• 2010

Some characterizations of regular $\Gamma$-rings are described by means of fuzzy ideals. The concepts of fuzzy interior ideals in $\Gamma$-rings and semisimple $\Gamma$-rings are introduced. Some characterizations of semisimple $\Gamma$-rings are investigated by means of fuzzy interior ideals.

• Journal of the Korean Mathematical Society
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• v.55 no.6
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• pp.1381-1388
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• 2018

Let ${\Delta}$ be a simplicial complex, $I_{\Delta}$ its Stanley-Reisner ideal and $K[{\Delta}]$ its Stanley-Reisner ring over a field K. Assume that ${\Gamma}(R)$ denotes the zero-divisor graph of a commutative ring R. Here, first we present a condition on two reduced Noetherian rings R and R', equivalent to ${\Gamma}(R){\cong}{\Gamma}(R{^{\prime}})$. In particular, we show that ${\Gamma}(K[{\Delta}]){\cong}{\Gamma}(K^{\prime}[{\Delta}^{\prime}])$ if and only if ${\mid}Ass(I_{\Delta}){\mid}={\mid}Ass(I_{{{\Delta}^{\prime}}}){\mid}$ and either ${\mid}K{\mid}$, ${\mid}K^{\prime}{\mid}{\leq}{\aleph}_0$ or ${\mid}K{\mid}={\mid}K^{\prime}{\mid}$. This shows that ${\Gamma}(K[{\Delta}])$ contains little information about $K[{\Delta}]$. Then, we define the squarefree zero-divisor graph of $K[{\Delta}]$, denoted by ${\Gamma}_{sf}(K[{\Delta}])$, and prove that ${\Gamma}_{sf}(K[{\Delta}){\cong}{\Gamma}_{sf}(K[{\Delta}^{\prime}])$ if and only if $K[{\Delta}]{\cong}K[{\Delta}^{\prime}]$. Moreover, we show how to find dim $K[{\Delta}]$ and ${\mid}Ass(K[{\Delta}]){\mid}$ from ${\Gamma}_{sf}(K[{\Delta}])$.

• Bulletin of the Korean Mathematical Society
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• v.56 no.3
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• pp.729-743
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• 2019

In this paper, we introduce strongly quasi primary ideals which is an intermediate class of primary ideals and quasi primary ideals. Let R be a commutative ring with nonzero identity and Q a proper ideal of R. Then Q is called strongly quasi primary if $ab{\in}Q$ for $a,b{\in}R$ implies either $a^2{\in}Q$ or $b^n{\in}Q$ ($a^n{\in}Q$ or $b^2{\in}Q$) for some $n{\in}{\mathbb{N}}$. We give many properties of strongly quasi primary ideals and investigate the relations between strongly quasi primary ideals and other classical ideals such as primary, 2-prime and quasi primary ideals. Among other results, we give a characterization of divided rings in terms of strongly quasi primary ideals. Also, we construct a subgraph of ideal based zero divisor graph ${\Gamma}_I(R)$ and denote it by ${\Gamma}^*_I(R)$, where I is an ideal of R. We investigate the relations between ${\Gamma}^*_I(R)$ and ${\Gamma}_I(R)$. Further, we use strongly quasi primary ideals and ${\Gamma}^*_I(R)$ to characterize von Neumann regular rings.

• Bulletin of the Korean Chemical Society
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• v.30 no.3
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• pp.691-694
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• 2009

Pauson-Khand reaction of hydroxyenynes with $Co_2(CO)_8$ in the presence of N-methylmorpholine N-oxide or trimethylamine N-oxide as a promoter produced polycyclic $gamma$-lactams as single stereoisomers in moderate to excellent yield. These are the first examples of an intramolecular Pauson-Khand reaction on a hydroxyenyne system tethered to a three- and four-membered ring as new skeletons with 5,6,5 fused ring systems.

• Journal of applied mathematics & informatics
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• v.8 no.3
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• pp.909-917
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• 2001

In this paper we define product between fuzzy ${H_v}-ideals$ of given ${H_v}-rings$. we consider the fundamental relation ${\gamma}^*$ defined on and ${H_v}-ring$ and give some properties of the fundamental relations and fundamental rings with respect to the product of fuzzy ${H_v}-ideals$.