• Kyungpook Mathematical Journal
• /
• v.59 no.4
• /
• pp.591-601
• /
• 2019

Let ℤ_n be the ring of integers modulo n and Γ(ℤ_n) the zero-divisor graph of ℤ_n. In this paper, we study some properties of Γ(ℤ_n). More precisely, we completely characterize the diameter and the girth of Γ(ℤ_n). We also calculate the chromatic number of Γ(ℤ_n).

• Kyungpook Mathematical Journal
• /
• v.46 no.2
• /
• pp.255-260
• /
• 2006

Let $m$ and $k$ be any positive integers, let $\mathbb{Z}_k$ the ring of integers of modulo $k$, let $G_m(\mathbb{Z}_k)$ the group of all $m$ by $m$ nonsingular matrices over $\mathbb{Z}_k$ and let ${\phi}_m(k)$ the order of $G_m(\mathbb{Z}_k)$. In this paper, ${\phi}_m(k)$ can be computed by the following investigation: First, for any relatively prime positive integers $s$ and $t$, $G_m(\mathbb{Z}_{st})$ is isomorphic to $G_m(\mathbb{Z}_s){\times}G_m(\mathbb{Z}_t)$. Secondly, for any positive integer $n$ and any prime $p$, ${\phi}_m(p^n)=p^{m^2}{\cdot}{\phi}_m(p^{n-1})=p{^{2m}}^2{\cdot}{\phi}_m(p^{n-2})={\cdots}=p^{{(n-1)m}^2}{\cdot}{\phi}_m(p)$, and so ${\phi}_m(k)={\phi}_m(p_1^n1){\cdot}{\phi}_m(p_2^{n2}){\cdots}{\phi}_m(p_s^{ns})$ for the prime factorization of $k$, $k=p_1^{n1}{\cdot}p_2^{n2}{\cdots}p_s^{ns}$.

• Korean Journal of Mathematics
• /
• v.27 no.3
• /
• pp.563-580
• /
• 2019

In this paper, we study the strong metric dimension of zero-divisor graph ${\Gamma}(R)$ associated to a ring R. This is done by transforming the problem into a more well-known problem of finding the vertex cover number ${\alpha}(G)$ of a strong resolving graph $G_{sr}$. We find the strong metric dimension of zero-divisor graphs of the ring ${\mathbb{Z}}_n$ of integers modulo n and the ring of Gaussian integers ${\mathbb{Z}}_n$[i] modulo n. We obtain the bounds for strong metric dimension of zero-divisor graphs and we also discuss the strong metric dimension of the Cartesian product of graphs.

• Journal of the Korean Mathematical Society
• /
• v.57 no.3
• /
• pp.571-583
• /
• 2020

The Galois ring R of characteristic pⁿ having p^mn elements is a finite extension of the ring of integers modulo pⁿ, where p is a prime number and n, m are positive integers. In this paper, we develop the concepts of Jacobi sums over R and under the assumption that the generating additive character of R is trivial on maximal ideal of R, we obtain the basic relationship between Gauss sums and Jacobi sums, which allows us to determine the absolute value of the Jacobi sums.

• Journal of the Korean Mathematical Society
• /
• v.47 no.4
• /
• pp.805-818
• /
• 2010

We classify linear maps which preserve idempotents on $n{\times}n$ matrices over some classes of semirings. Our results include many known semirings like the semiring of all nonnegative integers, the semiring of all nonnegative reals, any unital commutative ring, which is zero divisor free and of characteristic not two (not necessarily a principal ideal domain), and the ring of integers modulo m, where m is a product of distinct odd primes.

• Journal of applied mathematics & informatics
• /
• v.40 no.3_4
• /
• pp.729-740
• /
• 2022

Let ℤ be the ring of integers and let ℤ_n be the ring of integers modulo n. Let ℤ(+)ℤ_n be the idealization of ℤ_n in ℤ and let (ℤ(+)ℤ_n)[X]] be either (ℤ(+)ℤ_n)[X] or (ℤ(+)ℤ_n)[[X]]. In this article, we study the zero-divisor graphs of ℤ(+)ℤ_n and (ℤ(+)ℤ_n)[X]]. More precisely, we completely characterize the diameter and the girth of the zero-divisor graphs of ℤ(+)ℤ_n and (ℤ(+)ℤ_n)[X]]. We also calculate the chromatic number of the zero-divisor graphs of ℤ(+)ℤ_n and (ℤ(+)ℤ_n)[X]].

• East Asian mathematical journal
• /
• v.40 no.1
• /
• pp.95-107
• /
• 2024

Let K be an imaginary quadratic field with ring of integers 𝓞_K and N be a positive integer. By K_(N) we mean the ray class field of K modulo N𝓞_K. In this paper, for each prime p of K relatively prime to N𝓞_K we explicitly describe the action of the Artin symbol (${\frac{K_{(N)}/K}{p}}$) on special values of modular functions of level N. Furthermore, we extend the Kronecker congruence relation for the elliptic modular function j to some modular functions of higher level.

• Journal of the Chungcheong Mathematical Society
• /
• v.34 no.4
• /
• pp.317-326
• /
• 2021

Let K be an imaginary quadratic field other than ℚ(${\sqrt{-1}}$) and ℚ(${\sqrt{-3}}$), and let 𝒪_K be its ring of integers. Let N be a positive integer such that N = 5 or N ≥ 7. In this paper, we generate the ray class field modulo N𝒪_K over K by using a single x-coordinate of an elliptic curve with complex multiplication by 𝒪_K.

• Journal of the Korean Mathematical Society
• /
• v.55 no.2
• /
• pp.343-372
• /
• 2018

Let K be an imaginary quadratic field with ring of integers ${\mathcal{O}}_K$. Let E be an elliptic curve with complex multiplication by ${\mathcal{O}}_K$, and let $h_E$ be the Weber function on E. Let $N{\in}\{2,3,4,6\}$. We show that $h_E$ alone when evaluated at a certain N-torsion point on E generates the ray class field of K modulo $N{\mathcal{O}}_K$. This would be a partial answer to the question raised by Hasse and Ramachandra.

• Korean Journal of Mathematics
• /
• v.28 no.1
• /
• pp.65-73
• /
• 2020

This article concerns a property of local rings and domains. A ring R is called weakly local if for every a ∈ R, a is regular or 1-a is regular, where a regular element means a non-zero-divisor. We study the structure of weakly local rings in relation to several kinds of factor rings and ring extensions that play roles in ring theory. We prove that the characteristic of a weakly local ring is either zero or a power of a prime number. It is also shown that the weakly local property can go up to polynomial (power series) rings and a kind of Abelian matrix rings.