• Journal of the Korean Mathematical Society
• /
• v.57 no.1
• /
• pp.131-144
• /
• 2020

Characterizations of almost perfect domains by certain covers and envelopes, due to Bazzoni-Salce [7] and Bazzoni [4], are generalized to almost perfect commutative rings (with zero-divisors). These rings were introduced recently by Fuchs-Salce [14], showing that the new rings share numerous properties of the domain case. In this note, it is proved that admitting strongly flat covers characterizes the almost perfect rings within the class of commutative rings (Theorem 3.7). Also, the existence of projective dimension 1 covers characterizes the same class of rings within the class of commutative rings admitting the cotorsion pair (𝒫₁, 𝒟) (Theorem 4.1). Similar characterization is proved concerning the existence of divisible envelopes for h-local rings in the same class (Theorem 5.3). In addition, Bazzoni's characterization via direct sums of weak-injective modules [4] is extended to all commutative rings (Theorem 6.4). Several ideas of the proofs known for integral domains are adapted to rings with zero-divisors.

• Bulletin of the Korean Mathematical Society
• /
• v.58 no.6
• /
• pp.1387-1400
• /
• 2021

Suppose that $T=\(\array{A&0\\U&B}\)$ is a formal triangular matrix ring, where A and B are rings and U is a (B, A)-bimodule. Let ℭ₁ and ℭ₂ be two classes of left A-modules, 𝔇₁ and 𝔇₂ be two classes of left B-modules. We prove that (ℭ₁, ℭ₂) and (𝔇₁, 𝔇₂) are admissible balanced pairs if and only if (p(ℭ₁, 𝔇₁), h(ℭ₂, 𝔇₂) is an admissible balanced pair in T-Mod. Furthermore, we describe when ($P^{C_1}_{D_1}$, $I^{C_2}_{D_2}$) is an admissible balanced pair in T-Mod. As a consequence, we characterize when T is a left virtually Gorenstein ring.

• Journal of The Korean Astronomical Society
• /
• v.29 no.spc1
• /
• pp.45-48
• /
• 1996

Rings are common in disk galaxies. These rings are either indistinguishable from a pair of tightly wound spirals, or themselves are a part of the spiral structure. Furthermore, their occurrence is seen coincident with a bar in the center. In this paper, we interpret this spiral-ring structure as density waves resonantly excited by a rotating bar potential. The theory gives excellent agreement for the molecular spiral-rings in central parts. of nearby disk galaxies, observed by high resolution radio arrays. The same mechanism works for more distant spiral-rings in the outer parts of disk galaxies qualitatively, although the problem is complicated by the coupling of the stellar and gaseous disks.

• Kyungpook Mathematical Journal
• /
• v.57 no.2
• /
• pp.187-191
• /
• 2017

Let R be a domain with quotient field K and prime subring A. Then R is integral over each of its subrings having quotient field K if and only if (A, R) is a survival pair. This shows the redundancy of a condition involving going-down pairs in a earlier characterization of such rings. In characteristic 0, the domains being characterized are the rings R that are isomorphic to subrings of the ring of all algebraic integers. In positive (prime) characteristic, the domains R being characterized are of two kinds: either R = K is an algebraic field extension of A or precisely one valuation domain of K does not contain R.

• Kyungpook Mathematical Journal
• /
• v.51 no.1
• /
• pp.1-10
• /
• 2011

Let (R, T) be a normal pair of commutative rings (i.e., R ${\subseteq}$ T is a unita extension of commutative rings, not necessarily integral domains, such that S is integrally closed in T for each ring S such that R ${\subseteq}$ S ${\subseteq}$ T) such that the total quotient ring of R is a von Neumann regular ring. Let P be one of the following ring-theoretic properties: going-down ring, extensionally going-down (EGD) ring, locally divided ring. Then R has P if and only if T has P. An example shows that the "if" part of the assertion fails if P is taken to be the "divided domain" property.

• Bulletin of the Korean Mathematical Society
• /
• v.48 no.1
• /
• pp.157-167
• /
• 2011

For a ring endomorphism of a ring R, Krempa called $\alpha$ rigid endomorphism if $a{\alpha}(a)$ = 0 implies a = 0 for a $\in$ R, and Hong et al. called R an $\alpha$-rigid ring if there exists a rigid endomorphism $\alpha$. Due to Rege and Chhawchharia, a ring R is called Armendariz if whenever the product of any two polynomials in R[x] over R is zero, then so is the product of any pair of coefficients from the two polynomials. The Armendariz property of polynomials was extended to one of skew polynomials (i.e., $\alpha$-Armendariz rings and $\alpha$-skew Armendariz rings) by Hong et al. In this paper, we study the relationship between $\alpha$-rigid rings and extended Armendariz rings, and so we get various conditions on the rings which are equivalent to the condition of being an $\alpha$-rigid ring. Several known results relating to extended Armendariz rings can be obtained as corollaries of our results.

• Bulletin of the Korean Mathematical Society
• /
• v.60 no.2
• /
• pp.405-423
• /
• 2023

Let (𝓛, 𝒜) be a complete duality pair. We give some equivalent characterizations of Gorenstein (𝓛, 𝒜)-projective modules and construct some model structures associated to duality pairs and Frobenius pairs. Some rings are described by Frobenius pairs. In addition, we investigate special Gorenstein (𝓛, 𝒜)-projective modules and construct some model structures and recollements associated to them.

• Journal of the Korean Mathematical Society
• /
• v.55 no.6
• /
• pp.1389-1421
• /
• 2018

In this paper, we introduce the notion of C-Gorenstein weak injective modules with respect to a semidualizing bimodule $_SC_R$, where R and S are arbitrary associative rings. We show that an iteration of the procedure used to define $G_C$-weak injective modules yields exactly the $G_C$-weak injective modules, and then give the Foxby equivalence in this setting analogous to that of C-Gorenstein injective modules over commutative Noetherian rings. Finally, some applications are given, including weak co-Auslander-Buchweitz context, model structure and dual pair induced by $G_C$-weak injective modules.

• Communications of the Korean Mathematical Society
• /
• v.37 no.2
• /
• pp.337-345
• /
• 2022

Let R ⊆ T be an extension of a commutative ring and S ⊆ R a multiplicative subset. We say that (R, T) is an S-accr (a commutative ring R is said to be S-accr if every ascending chain of residuals of the form (I : B) ⊆ (I : B²) ⊆ (I : B³) ⊆ ⋯ is S-stationary, where I is an ideal of R and B is a finitely generated ideal of R) pair if every ring A with R ⊆ A ⊆ T satisfies S-accr. Using this concept, we give an S-version of several different known results.

• Bulletin of the Korean Mathematical Society
• /
• v.46 no.5
• /
• pp.857-866
• /
• 2009

Let n be a fixed positive integer, R be a 2n!-torsion free prime ring and $\mu$, $\nu$ be a pair of generalized derivations on R. If < $\mu^2(x)+\nu(x),\;x^n$ > = 0 for all x $\in$ R, then $\mu$ and $\nu$ are either left multipliers or right multipliers. Let n be a fixed positive integer, R be a noncommutative 2n!-torsion free prime ring with the center $C_R$ and d, g be a pair of derivations on R. If < $d^2(x)+g(x)$, $x^n$ > $\in$ $C_R$ for all x $\in$ R, then d = g = 0. Then we apply these purely algebraic techniques to obtain several range inclusion results of pair of (generalized-)derivations on a Banach algebra.