• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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• pp.1189-1208
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• 2018

The aim of this paper is to study the class of ${\Lambda}$-constacyclic codes of length $2p^s$ over the finite commutative chain ring ${\mathcal{R}}_a=\frac{{\mathbb{F}_{p^m}}[u]}{{\langle}u^a{\rangle}}={\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+{\cdots}+u^{a-1}{\mathbb{F}}_{p^m}$, for all units ${\Lambda}$ of ${\mathcal{R}}_a$ that have the form ${\Lambda}={\Lambda}_0+u{\Lambda}_1+{\cdots}+u^{a-1}{\Lambda}_{a-1}$, where ${\Lambda}_0,{\Lambda}_1,{\cdots},{\Lambda}_{a-1}{\in}{\mathbb{F}}_{p^m}$, ${\Lambda}_0{\neq}0$, ${\Lambda}_1{\neq}0$. The algebraic structure of all ${\Lambda}$-constacyclic codes of length $2p^s$ over ${\mathcal{R}}_a$ and their duals are established. As an application, this structure is used to determine the Rosenbloom-Tsfasman (RT) distance and weight distributions of all such codes. Among such constacyclic codes, the unique MDS code with respect to the RT distance is obtained.

• Bulletin of the Korean Mathematical Society
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• v.29 no.1
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• pp.65-72
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• 1992

All rings are commutative, Noetherian with identity and of prime characteristic p, unless otherwise specified. First, we describe the definition of tight closure of an ideal and the properties about the tight closure used frequently. The technique used here for the tight closure was introduced by M. Hochster and C. Huneke [4,5, or 6]. Using the concepts of the tight closure and its properties, we will prove that if R is a complete local domain and F-rational, then R is Cohen-Macaulay. Next, we study the properties of R$^{+}$, the integral closure of a domain in an algebraic closure of its field of fractions. In fact, if R is a complete local domain of characteristic p>0, then R$^{+}$ is Cohen-Macaulay [8]. But we do not know this fact is true or not if the characteristic of R is zero. For the special case we can show that if R is a non-Cohen-Macaulay normal domain containing the rationals Q, then R$^{+}$ is not Cohen-Macaulay. Finally we will prove that if R is an excellent local domain of characteristic p and F-ratiional, then R is Cohen-Macaulay.aulay.

• Bulletin of the Korean Mathematical Society
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• v.55 no.2
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• pp.625-631
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• 2018

Let R be a commutative ring with identity and let R[x] be the collection of polynomials with coefficients in R. There are a lot of multiplications in R[x] such that together with the usual addition, R[x] becomes a ring that contains R as a subring. These multiplications are from a class of functions ${\lambda}$ from ${\mathbb{N}}_0$ to ${\mathbb{N}}$. The trivial case when ${\lambda}(i)=1$ for all i gives the usual polynomial ring. Among nontrivial cases, there is an important one, namely, the case when ${\lambda}(i)=i!$ for all i. For this case, it gives the well-known Hurwitz polynomial ring $R_H[x]$. In this paper, we completely determine the Krull dimension of $R_H[x]$ when R is a $Pr{\ddot{u}}fer$ domain. Let R be a $Pr{\ddot{u}}fer$ domain. We show that dim $R_H[x]={\dim}\;R+1$ if R has characteristic zero and dim $R_H[x]={\dim}\;R$ otherwise.

• Journal of applied mathematics & informatics
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• v.5 no.3
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• pp.819-826
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• 1998

The purpose of this paper is to prove the following results; (1) Let R be a prime ring of char $(R)\neq 2$ and I a nonzero left ideal of R. The existence of a nonzero symmetric bi-derivation D : $R\timesR\;\longrightarrow\;$ such that d is sew-commuting on I where d is the trace of D forces R to be commutative (2) Let m and n be integers with $m\;\neq\;0.\;or\;n\neq\;0$. Let R be a noncommutative prime ring of char$ (R))\neq \; 2-1\; p_1 \;n_1$ where p is a prime number which is a divisor of m, and I a nonzero two-sided ideal of R. Let $D_1$ ; $R\;\times\;R\;\longrightarrow\;and\;$ $D_2\;:\;R\;\times\;R\;longrightarrow\;R$ be symmetric bi-derivations. Suppose further that there exists a symmetric bi-additive mapping B ; $R\;\times\;R\;\longrightarrow\;and\;$ such that $md_1(\chi)\chi ＋ n\chi d_2(\chi)=f(\chi$) holds for all $\chi$$\in$I, where $d_1 \;and\; d_2$ are the traces of $D_1 \;and\; D_2$ respectively and f is the trace of B. Then we have $D_1=0 \;and\; D_2=0$.

• Journal of the Korean Mathematical Society
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• v.45 no.2
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• pp.425-433
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• 2008

Let R be a ring with identity $1_R$ and let U(R) denote the group of all units of R. A ring R is called locally finite if every finite subset in it generates a finite semi group multiplicatively. In this paper, some results are obtained as follows: (1) for any semilocal (hence semiperfect) ring R, U(R) is a finite (resp. locally finite) group if and only if R is a finite (resp. locally finite) ring; U(R) is a locally finite group if and only if U$(M_n(R))$ is a locally finite group where $M_n(R)$ is the full matrix ring of $n{\times}n$ matrices over R for any positive integer n; in addition, if $2=1_R+1_R$ is a unit in R, then U(R) is an abelian group if and only if R is a commutative ring; (2) for any semiperfect ring R, if E(R), the set of all idempotents in R, is commuting, then $R/J\cong\oplus_{i=1}^mD_i$ where each $D_i$ is a division ring for some positive integer m and |E(R)|=$2^m$; in addition, if 2=$1_R+1_R$ is a unit in R, then every idempotent is central.

• Bulletin of the Korean Mathematical Society
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• v.20 no.1
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• pp.45-49
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• 1983

If R is ring and M is a right (or left) R-module, then M is called a faithful R-module if, for some a in R, x.a=0 for all x.mem.M then a=0. In [4], R.E. Johnson defines that M is a prime module if every non-zero submodule of M is faithful. Let us define that M is of prime type provided that M is faithful if and only if every non-zero submodule is faithful. We call a right (left) ideal I of R is of prime type if R/I is of prime type as a R-module. This is equivalent to the condition that if xRy.subeq.I then either x.mem.I ro y.mem.I (see [5:3:1]). It is easy to see that in case R is a commutative ring then a right or left ideal of a prime type is just a prime ideal. We have defined in [5], that a chain of right ideals of prime type in a ring R is a finite strictly increasing sequence I$_{0}$.contnd.I$_{1}$.contnd....contnd.I$_{n}$; the length of the chain is n. By the right dimension of a ring R, which is denoted by dim, R, we mean the supremum of the length of all chains of right ideals of prime type in R. It is an integer .geq.0 or .inf.. The left dimension of R, which is denoted by dim$_{l}$ R is similarly defined. It was shown in [5], that dim$_{r}$R=0 if and only if dim$_{l}$ R=0 if and only if R modulo the prime radical is a strongly regular ring. By "a strongly regular ring", we mean that for every a in R there is x in R such that axa=a=a$^{2}$x. It was also shown that R is a simple ring if and only if every right ideal is of prime type if and only if every left ideal is of prime type. In case, R is a (right or left) primitive ring then dim$_{r}$R=n if and only if dim$_{l}$ R=n if and only if R.iden.D$_{n+1}$ , n+1 by n+1 matrix ring on a division ring D. in this paper, we establish the following results: (1) If R is prime ring and dim$_{r}$R=n then either R is a righe Ore domain such that every non-zero right ideal of a prime type contains a non-zero minimal prime ideal or the classical ring of ritght quotients is isomorphic to m*m matrix ring over a division ring where m.leq.n+1. (b) If R is prime ring and dim$_{r}$R=n then dim$_{l}$ R=n if dim$_{l}$ R=n if dim$_{l}$ R<.inf. (c) Let R be a principal right and left ideal domain. If dim$_{r}$R=1 then R is an unique factorization domain.TEX>R=1 then R is an unique factorization domain.