• Journal of the Korean Mathematical Society
• /
• v.59 no.4
• /
• pp.821-841
• /
• 2022

In this paper, we introduce and study regular rings relative to the hereditary torsion theory w (a special case of a well-centered torsion theory over a commutative ring), called w-regular rings. We focus mainly on the w-regularity for w-coherent rings and w-Noetherian rings. In particular, it is shown that the w-coherent w-regular domains are exactly the Prüfer v-multiplication domains and that an integral domain is w-Noetherian and w-regular if and only if it is a Krull domain. We also prove the w-analogue of the global version of the Serre-Auslander-Buchsbaum Theorem. Among other things, we show that every w-Noetherian w-regular ring is the direct sum of a finite number of Krull domains. Finally, we obtain that the global weak w-projective dimension of a w-Noetherian ring is 0, 1, or ∞.

• Journal of the Korean Mathematical Society
• /
• v.60 no.2
• /
• pp.407-464
• /
• 2023

Let R be a commutative ring with identity. The structure theorem says that R is a PIR (resp., UFR, general ZPI-ring, π-ring) if and only if R is a finite direct product of PIDs (resp., UFDs, Dedekind domains, π-domains) and special primary rings. All of these four types of integral domains are Krull domains, so motivated by the structure theorem, we study the prime factorization of ideals in a ring that is a finite direct product of Krull domains and special primary rings. Such a ring will be called a general Krull ring. It is known that Krull domains can be characterized by the star operations v or t as follows: An integral domain R is a Krull domain if and only if every nonzero proper principal ideal of R can be written as a finite v- or t-product of prime ideals. However, this is not true for general Krull rings. In this paper, we introduce a new star operation u on R, so that R is a general Krull ring if and only if every proper principal ideal of R can be written as a finite u-product of prime ideals. We also study several ring-theoretic properties of general Krull rings including Kaplansky-type theorem, Mori-Nagata theorem, Nagata rings, and Noetherian property.

• Korean Journal of Mathematics
• /
• v.21 no.1
• /
• pp.23-28
• /
• 2013

The structures of finite local rings of order ${\leq}$ 16 with nonzero Jacobson radical are investigated. The whole shape of non-commutative local rings of minimal order is completely determined up to isomorphism.

• Bulletin of the Korean Mathematical Society
• /
• v.30 no.1
• /
• pp.61-64
• /
• 1993

In this paper, we show that if R is a local-global domain then the Question holds. McDonald and Waterhouse in [6] and Estes and Guralnick in [5] introduced the concept of local-global rings (so called rings with many units) independently. A local-global ring is a commutative ring R with 1 satisfying; if a polynomial f in R[ $x_{1}$, .., $x_{n}$] represents a unit over $R_{P}$ for every maximal ideal P in R, then f represents a unit over R. Such rings include semilocal rings, or more generally, rings which are von Neumann regular mod their Jacobson radical, and the ring of all algebraic integers.s.s.

• Journal of the Korean Mathematical Society
• /
• v.51 no.5
• /
• pp.971-985
• /
• 2014

All modules considered in this note are over associative commutative rings with an identity element. We show that a ${\omega}$-local module M is Rad-supplemented if and only if M/P(M) is a local module, where P(M) is the sum of all radical submodules of M. We prove that ${\omega}$-local nonsmall submodules of a cyclic Rad-supplemented module are again Rad-supplemented. It is shown that commutative Noetherian rings over which every w-local Rad-supplemented module is supplemented are Artinian. We also prove that if a finitely generated Rad-supplemented module is cyclic or multiplication, then it is amply Rad-supplemented. We conclude the paper with a characterization of finitely generated amply Rad-supplemented left modules over any ring (not necessarily commutative).

• Kyungpook Mathematical Journal
• /
• v.61 no.4
• /
• pp.711-725
• /
• 2021

Let R be a commutative ring with nonzero identity, 𝓘(𝓡) the set of all ideals of R and δ : 𝓘(𝓡) → 𝓘(𝓡) an expansion of ideals of R. In this paper, we introduce the concept of 2-absorbing δ-semiprimary ideals in commutative rings which is an extension of 2-absorbing ideals. A proper ideal I of R is called 2-absorbing δ-semiprimary ideal if whenever a, b, c ∈ R and abc ∈ I, then either ab ∈ δ(I) or bc ∈ δ(I) or ac ∈ δ(I). Many properties and characterizations of 2-absorbing δ-semiprimary ideals are obtained. Furthermore, 2-absorbing δ₁-semiprimary avoidance theorem is proved.

• Journal of the Korean Mathematical Society
• /
• v.42 no.6
• /
• pp.1153-1167
• /
• 2005

In this paper, all rings and all near-rings R are associative, all modules are right R-modules. For a near-ring R, we consider representations of R as R-groups. We start with a study of AGR rings and their properties. Next, for any right R-module M, we define a new concept GM module and investigate the commutative property of faithful GM modules and some characterizations of GM modules. Similarly, for any near-ring R, we introduce an R-group with MR-property and some properties of MR groups.

• Journal of the Chungcheong Mathematical Society
• /
• v.30 no.3
• /
• pp.313-321
• /
• 2017

In this paper, we introduce the notion of a ${\ast}$-semiderivation on ${\ast}$-rings, and we try to extend some results for derivations of rings or near-rings to a more general case for ${\ast}$-semiderivations of prime ${\ast}$-rings.

• Bulletin of the Korean Mathematical Society
• /
• v.39 no.3
• /
• pp.411-422
• /
• 2002

In this note we are concerned with relationships between one-sided ideals and two-sided ideals, and study the properties of polynomial rings whose maximal one-sided ideals are two-sided, in the viewpoint of the Nullstellensatz on noncommutative rings. Let R be a ring and R[x] be the polynomial ring over R with x the indeterminate. We show that eRe is right quasi-duo for $0{\neq}e^2=e{\in}R$ if R is right quasi-duo; R/J(R) is commutative with J(R) the Jacobson radical of R if R[$\chi$] is right quasi-duo, from which we may characterize polynomial rings whose maximal one-sided ideals are two-sided; if R[x] is right quasi-duo then the Jacobson radical of R[x] is N(R)[x] and so the $K\ddot{o}the's$ conjecture (i.e., the upper nilradical contains every nil left ideal) holds, where N(R) is the set of all nilpotent elements in R. Next we prove that if the polynomial rins R[x], over a reduced ring R with $\mid$X$\mid$ $\geq$ 2, is right quasi-duo, then R is commutative. Several counterexamples are included for the situations that occur naturally in the process of this note.

• Bulletin of the Korean Mathematical Society
• /
• v.52 no.1
• /
• pp.125-135
• /
• 2015

Let R be a commutative ring and U an R-module. The aim of this paper is to study the duality between U-reflexive (pre)envelopes and U-reflexive (pre)covers of R-modules.