• 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.25 no.4
• /
• pp.771-777
• /
• 2012

We give a result on the density of certain additive arithmetic functions which count the number of restricted irreducible polynomials on the polynomial ring.

• Communications of the Korean Mathematical Society
• /
• v.38 no.1
• /
• pp.69-77
• /
• 2023

In this article we relate the six Prüfer conditions with the EM conditions. We use the EM-conditions to prove some cases of equivalence of the six Prüfer conditions. We also use the Prüfer conditions to answer some open problems concerning EM-rings.

• 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.

• Communications of the Korean Mathematical Society
• /
• v.34 no.2
• /
• pp.361-374
• /
• 2019

Let R be a ring, ($S,{\leq}$) a strictly ordered monoid, and ${\omega}:S{\rightarrow}End(R)$ a monoid homomorphism. In [18], Mazurek, and Ziembowski investigated when the skew generalized power series ring $R[[S,{\omega}]]$ is a domain satisfying the ascending chain condition on principal left (resp. right) ideals. Following [18], we obtain necessary and sufficient conditions on R, S and ${\omega}$ such that the skew generalized power series ring $R[[S,{\omega}]]$ is a right or left Archimedean domain. As particular cases of our general results we obtain new theorems on the ring of arithmetical functions and the ring of generalized power series. Our results extend and unify many existing results.