• Communications of the Korean Mathematical Society
• /
• v.10 no.4
• /
• pp.783-787
• /
• 1995

In this paper we characterize cancellation ideals in terms of multiplication ideals. Especially, we find a condition for an ideal generated by three elements to be a cancellation ideal.

• Journal of the Chungcheong Mathematical Society
• /
• v.25 no.4
• /
• pp.717-723
• /
• 2012

In this paper, we investigate when gr-multiplication modules are finitely generated and show that if M is a finitely generated gr-multiplication R-module then there is a lattice isomorphism between the lattice of all graded ideals I of R containing ann(M) and the lattice of all graded submodules of M.

• Honam Mathematical Journal
• /
• v.31 no.3
• /
• pp.429-436
• /
• 2009

In this short paper, we prove some new properties on homogeneous ideals and primary ideals of R(+)M.

• Bulletin of the Korean Mathematical Society
• /
• v.32 no.2
• /
• pp.321-328
• /
• 1995

In this note all are commutative rings with identity and all modules are unital. Let R be a ring. An R-module M is called a multiplication module if for every submodule N of M there esists an ideal I of R such that N = IM. Clearly the ring R is a multiplication module as a module over itself. Also, it is well known that invertible and more generally profective ideals of R are multiplication R-modules (see [11, Theorem 1]).

• Bulletin of the Korean Mathematical Society
• /
• v.59 no.2
• /
• pp.397-405
• /
• 2022

In this article we investigate the transfer of multiplication-like properties to homomorphic images, direct products and amalgamated duplication of a ring along an ideal. Our aim is to provide examples of new classes of commutative rings satisfying the above-mentioned properties.

• Bulletin of the Korean Mathematical Society
• /
• v.47 no.5
• /
• pp.1053-1066
• /
• 2010

In this paper, several properties of endomorphism rings of modules are investigated. A multiplication module M over a commutative ring R induces a commutative ring $M^*$ of endomorphisms of M and hence the relation between the prime (maximal) submodules of M and the prime (maximal) ideals of $M^*$ can be found. In particular, two classes of ideals of $M^*$ are discussed in this paper: one is of the form $G_{M^*}\;(M,\;N)\;=\;\{f\;{\in}\;M^*\;|\;f(M)\;{\subseteq}\;N\}$ and the other is of the form $G_{M^*}\;(N,\;0)\;=\;\{f\;{\in}\;M^*\;|\;f(N)\;=\;0\}$ for a submodule N of M.

• Journal of the Korean Mathematical Society
• /
• v.60 no.2
• /
• pp.327-339
• /
• 2023

Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. In this article we introduce a new class of ring, called S-multiplication rings which are S-versions of multiplication rings. An R-module M is said to be S-multiplication if for each submodule N of M, sN ⊆ JM ⊆ N for some s ∈ S and ideal J of R (see for instance [4, Definition 1]). An ideal I of R is called S-multiplication if I is an S-multiplication R-module. A commutative ring R is called an S-multiplication ring if each ideal of R is S-multiplication. We characterize some special rings such as multiplication rings, almost multiplication rings, arithmetical ring, and S-P IR. Moreover, we generalize some properties of multiplication rings to S-multiplication rings and we study the transfer of this notion to various context of commutative ring extensions such as trivial ring extensions and amalgamated algebras along an ideal.

• Journal of the Chungcheong Mathematical Society
• /
• v.4 no.1
• /
• pp.99-102
• /
• 1991

Let R be a commutative ring with identity. A ring is said to be a regular multiplication ring if $A{\subseteq}B$, where A and B are ideals of R with B regular, implies that there exists an ideal C of R such that A = BC. We characterize such rings and study their properties.

• Korean Journal of Mathematics
• /
• v.18 no.4
• /
• pp.335-342
• /
• 2010

Let D be an integral domain with quotient field K,$\mathcal{I}(D)$ be the set of nonzero ideals of D, and $w$ be the star-operation on D defined by $I_w=\{x{\in}K{\mid}xJ{\subseteq}I$ for some $J{\in}\mathcal{I}(D)$ such that J is finitely generated and $J^{-1}=D\}$. The D is called a Pr$\ddot{u}$fer $v$-multiplication domain if $(II^{-1})_w=D$ for all nonzero finitely generated ideals I of D. In this paper, we show that D is a Pr$\ddot{u}$fer $v$-multiplication domain if and only if $(A{\cap}(B+C))_w=((A{\cap}B)+(A{\cap}C))_w$ for all $A,B,C{\in}\mathcal{I}(D)$, if and only if $(A(B{\cap}C))_w=(AB{\cap}AC)_w$ for all $A,B,C{\in}\mathcal{I}(D)$, if and only if $((A+B)(A{\cap}B))_w=(AB)_w$ for all $A,B{\in}\mathcal{I}(D)$, if and only if $((A+B):C)_w=((A:C)+(B:C))_w$ for all $A,B,C{\in}\mathcal{I}(D)$ with C finitely generated, if and only if $((a:b)+(b:a))_w=D$ for all nonzero $a,b{\in}D$, if and only if $(A:(B{\cap}C))_w=((A:B)+(A:C))_w$ for all $A,B,C{\in}\mathcal{I}(D)$ with B, C finitely generated.

• Journal of the Chungcheong Mathematical Society
• /
• v.13 no.2
• /
• pp.9-17
• /
• 2001

A ring R is called right mininjective if every isomorphsim between simple right ideals is given by left multiplication by an element of R. In this paper we consider that the necessary and sufficient condition for that Trivial extension of R by V, i.e. T(R; V ) is mininjective. We also study the split null extension R and S by V.