• 대한수학회보
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• 제52권3호
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• pp.935-946
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• 2015

The purpose of this paper is to introduce a new class of rings that is closely related to the classes of pseudo valuation rings (PVRs) and pseudo-almost valuation domains (PAVDs). A commutative ring R is said to be ${\phi}$-ring if its nilradical Nil(R) is both prime and comparable with each principal ideal. The name is derived from the natural map ${\phi}$ from the total quotient ring T(R) to R localized at Nil(R). A prime ideal P of a ${\phi}$-ring R is said to be a ${\phi}$-pseudo-strongly prime ideal if, whenever $x,y{\in}R_{Nil(R)}$ and $(xy){\phi}(P){\subseteq}{\phi}(P)$, then there exists an integer $m{\geqslant}1$ such that either $x^m{\in}{\phi}(R)$ or $y^m{\phi}(P){\subseteq}{\phi}(P)$. If each prime ideal of R is a ${\phi}$-pseudo strongly prime ideal, then we say that R is a ${\phi}$-pseudo-almost valuation ring (${\phi}$-PAVR). Among the properties of ${\phi}$-PAVRs, we show that a quasilocal ${\phi}$-ring R with regular maximal ideal M is a ${\phi}$-PAVR if and only if V = (M : M) is a ${\phi}$-almost chained ring with maximal ideal $\sqrt{MV}$. We also investigate the overrings of a ${\phi}$-PAVR.

• 대한수학회논문집
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• 제31권2호
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• pp.237-246
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• 2016

The purpose of this paper is to introduce some new class of rings that are closely related to the classes of sharp domains, pseudo-Dededkind domains, TV domains and finite character domains. A ring R is called a ${\phi}$-sharp ring if whenever for nonnil ideals I, A, B of R with $I{\supseteq}AB$, then I = A'B' for nonnil ideals A', B' of R where $A^{\prime}{\supseteq}A$ and $B^{\prime}{\supseteq}B$. We proof that a ${\phi}$-Dedekind ring is a ${\phi}$-sharp ring and we get some properties that by them a ${\phi}$-sharp ring is a ${\phi}$-Dedekind ring.

• 대한수학회보
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• 제22권1호
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• pp.1-5
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• 1985

The differential polynomial ring A[X, D] has been studied by many authors J. Cozzens, C. Faith, R.E. Johnson and D. Mathis and others. The main purpose of the present paper is to study some properties of .phi.-differential polynomial ring A[X, D, .phi.] and modules over the .phi.-differential polynomial ring X[X, D, .phi.].

• 대한수학회지
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• 제53권5호
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• pp.1057-1075
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• 2016

Let R be a ring in which Nil(R) is a divided prime ideal of R. Then, for a suitable property X of integral domains, we can define a ${\phi}$-X-ring if R/Nil(R) is an X-domain. This device was introduced by Badawi [8] to study rings with zero divisors with a homomorphic image a particular type of domain. We use it to introduce and study a number of concepts such as ${\phi}$-Schreier rings, ${\phi}$-quasi-Schreier rings, ${\phi}$-almost-rings, ${\phi}$-almost-quasi-Schreier rings, ${\phi}$-GCD rings, ${\phi}$-generalized GCD rings and ${\phi}$-almost GCD rings as rings R with Nil(R) a divided prime ideal of R such that R/Nil(R) is a Schreier domain, quasi-Schreier domain, almost domain, almost-quasi-Schreier domain, GCD domain, generalized GCD domain and almost GCD domain, respectively. We study some generalizations of these concepts, in light of generalizations of these concepts in the domain case, as well. Here a domain D is pre-Schreier if for all $x,y,z{\in}D{\backslash}0$, x | yz in D implies that x = rs where r | y and s | z. An integrally closed pre-Schreier domain was initially called a Schreier domain by Cohn in [15] where it was shown that a GCD domain is a Schreier domain.

• 대한수학회지
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• 제55권5호
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• pp.1221-1233
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• 2018

Let R be a commutative ring with non-zero identity and let NN(R) = {I | I is a nonnil ideal of R}. Let M be an R-module and let ${\phi}-tor(M)=\{x{\in}M{\mid}Ix=0\text{ for some }I{\in}NN(R)\}$. If ${\phi}or(M)=M$, then M is called a ${\phi}$-torsion module. An R-module M is said to be ${\phi}$-flat, if $0{\rightarrow}{A{\otimes}_R}\;{M{\rightarrow}B{\otimes}_R}\;{M{\rightarrow}C{\otimes}_R}\;M{\rightarrow}0$ is an exact R-sequence, for any exact sequence of R-modules $0{\rightarrow}A{\rightarrow}B{\rightarrow}C{\rightarrow}0$, where C is ${\phi}$-torsion. In this paper, the concepts of NRD-submodules and NP-submodules are introduced, and the ${\phi}$-flat modules over a ${\phi}-Pr{\ddot{u}}fer$ ring are investigated.

• 대한수학회지
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• 제50권1호
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• pp.219-229
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• 2013

Let R be a commutative ring with $1{\neq}0$ and let $\mathcal{H}$ = {R|R is a commutative ring and Nil(R) is a divided prime ideal}. If $R{\in}\mathcal{H}$, then R is called a ${\phi}$-ring. In this paper, we introduce the concepts of ${\phi}$-torsion modules, ${\phi}$-flat modules, and ${\phi}$-von Neumann regular rings.

• 대한수학회보
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• 제58권4호
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• pp.1039-1052
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• 2021

In this paper, we introduce and study the class of 𝜙-w-flat modules which are generalizations of both 𝜙-flat modules and w-flat modules. The 𝜙-w-weak global dimension 𝜙-w-w.gl.dim(R) of a commutative ring R is also introduced and studied. We show that, for a 𝜙-ring R, 𝜙-w-w.gl.dim(R) = 0 if and only if w-dim(R) = 0 if and only if R is a 𝜙-von Neumann ring. It is also proved that, for a strongly 𝜙-ring R, 𝜙-w-w.gl.dim(R) ≤ 1 if and only if each nonnil ideal of R is 𝜙-w-flat, if and only if R is a 𝜙-PvMR, if and only if R is a PvMR.

• Kyungpook Mathematical Journal
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• 제46권2호
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• pp.255-260
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• 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 $k$, $k=p_1^{n1}{\cdot}p_2^{n2}{\cdots}p_s^{ns}$.

• 열처리공학회지
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• 제17권1호
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• pp.10-16
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• 2004

The aim of this study is to investigate the process conditions on the microstructual changes and mechanical properties of large 7175 aluminum ring roll forgings. The billets range from 370 mm to 720 mm in diameter were homogenized and ring roll forged after direct chill casting. The tensile properties of ring roll forged specimen manufactured with ${\Phi}370mm$ billets were superior to those of ${\Phi}720mm$ billets under $T_6$ condition. Also, these properties showed better than those of military specification under $T_{74}$ treatment. The impact value of ring roll forged specimen under $T_{74}$ treatment increased up to 20% than that of $T_6$ condition. The fracture toughness of ring roll forged specimen manufactured with ${\Phi}370mm$ cast billet showed nearly same level of ${\Phi}720mm$ billet which was processed using MF or Cog free forging followed by ring roll forging.

• 대한수학회보
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• 제22권1호
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• pp.11-15
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• 1985

Manddelberg [9] has shown that a Clifford algebra of a free quadratic space over an arbitrary semi-local ring R in Brawer-Wall group BW(R) is determined by its rank, determinant, and Hasse invariant. In this paper, we prove a corresponding result when R is a full ring.Throughout this paper, unless otherwise specified, we assume that R is a commutative ring having 2 a unit. A quadratic space (V, B, .phi.) over R is a finitely generated projective R-module V with a symmetric bilinear mapping B: V*V.rarw.R which is non-degenerate (i.e., the natural mapping V.rarw.Ho $m_{R}$(V,R) induced by B is an isomorphism), and with a quadratic mapping .phi.: V.rarw.R such that B(x,y)=1/2(.phi.(x+y)-.phi.(x)-.phi.(y)) and .phi.(rx) = $r^{2}$.phi.(x) for all x, y in V and r in R. We denote the group of multiplicative units of R by U9R). If (V, B, .phi.) is a free rank n quadratic space over R with an orthogonal basis { $x_{1}$,.., $x_{n}$}, we will write < $a_{1}$,.., $a_{n}$> for (V, B, .phi.) where the $a_{i}$=.phi.( $x_{i}$) are in U(R), and denote the space by the table [ $a_{ij}$ ] where $a_{ij}$ =B( $x_{i}$, $x_{j}$). In the case n=2 and B( $x_{1}$, $x_{2}$)=1/2 we reserve the notation [a $a_{11}$, $a_{22}$] for the space. A commutative ring R having 2 a unit is called full [10] if for every triple $a_{1}$, $a_{2}$, $a_{3}$ of elements in R with ( $a_{1}$, $a_{2}$, $a_{3}$)=R, there is an element w in R such that $a_{1}$+ $a_{2}$w+ $a_{3}$ $w^{2}$=unit.TEX>=unit.t.t.t.