• Journal of applied mathematics & informatics
• /
• v.9 no.2
• /
• pp.851-857
• /
• 2002

In this paper, we introduce a congruence relation on a seminear-ring and study quotient structures on it. Also, we investigate homomorphisms on a seminear-ring.

• Journal of applied mathematics & informatics
• /
• v.8 no.3
• /
• pp.855-867
• /
• 2001

This note we give a construction of a quotient ring $R/{\mu}$ induced via a fuzzy ideal ${\mu}$ in a ring R. The Fuzzy First, Second and Third Isomorphism Theorems are established. For some applications of this construction of quotient rings, we show that if ${\mu}$ is a fuzzy ideal of a commutative ring R, then $\mu$ is prime (resp. $R/{\mu}$ is a field, every zero divisor in $R/{\mu}$ is nilpotent). Moreover we give a simpler characterization of fuzzy maximal ideal of a ring.

• Journal of applied mathematics & informatics
• /
• v.7 no.1
• /
• pp.289-295
• /
• 2000

In this note, we define a ${Q^*}-ideal$ in a seminear-ring which is analogous of a Q-ideal in a semiring, and we construct a quotient seminear-ring. Also, We prove the fundamental theorem of homomorphisms for seminear-rings.

• 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.46 no.1
• /
• pp.135-146
• /
• 2009

In the present note we study the properties of weak Armendariz rings, and the connections among weak Armendariz rings, Armendariz rings, reduced rings and IFP rings. We prove that a right Ore ring R is weak Armendariz if and only if so is Q, where Q is the classical right quotient ring of R. With the help of this result we can show that a semiprime right Goldie ring R is weak Armendariz if and only if R is Armendariz if and only if R is reduced if and only if R is IFP if and only if Q is a finite direct product of division rings, obtaining a simpler proof of Lee and Wong's result. In the process we construct a semiprime ring extension that is infinite dimensional, from given any semi prime ring. We next find more examples of weak Armendariz rings.

• The Pure and Applied Mathematics
• /
• v.7 no.1
• /
• pp.33-40
• /
• 2000

We prove some characterization of rings with chain conditions in terms of fuzzy quotient rings and fuzzy ideals. We also show that a ring R is left Artinian if and only of the set of values of every fuzzy ideal on R is upper well-ordered.

• Communications of the Korean Mathematical Society
• /
• v.32 no.4
• /
• pp.789-803
• /
• 2017

Let $K={\mathbb{Q}}({\alpha})$ be a cubic field where ${\alpha}$ is an algebraic integer such that $disc_K({\alpha})$ is square-free. In this paper we will classify the structure of the unit group of the quotient ring ${\mathcal{O}}_K/A$ for each non-zero ideal A of ${\mathcal{O}}_K$.

• Bulletin of the Korean Mathematical Society
• /
• v.61 no.1
• /
• pp.147-160
• /
• 2024

In this paper, we define two new classes of monomial ideals I_𝑙,d and J_k,d. When d ≥ 2k + 1 and 𝑙 ≤ d - k - 1, we give the exact formulas to compute the depth and Stanley depth of quotient rings S/I^t_𝑙,d for all t ≥ 1. When d = 2k = 2𝑙, we compute the depth and Stanley depth of quotient ring S/I_𝑙,d. When d ≥ 2k, we also compute the depth and Stanley depth of quotient ring S/J_k,d.

• Communications of the Korean Mathematical Society
• /
• v.37 no.1
• /
• pp.57-64
• /
• 2022

In this paper, we show, among others, that if A is a Banach algebra satisfying a functional identity involving a b-generalized derivation F on A, under some mild conditions, is of the form F(x) = ax for all x ∈ R, where a ∈ Q_r, a right Martindale quotient ring of A.

• Communications of the Korean Mathematical Society
• /
• v.19 no.2
• /
• pp.233-242
• /
• 2004

The aim of this note is to study properties of the generalized centroid of the semi-prime gamma rings. Main results are the following theorems: (1) Let M be a semi-prime $\Gamma$-ring and Q a quotient $\Gamma$-ring of M. If W is a non-zero submodule of the right (left) M-module Q, then $W\Gamma$W $\neq 0. Furthermore Q is a semi-prime $\Gamma$-ring. (2) Let M be a semi-prime $\Gamma$-ring and $C_{{Gamma}$ the generalized centroid of M. Then $C_{\Gamma}$ is a regular $\Gamma$-ring. (3) Let M be a semi-prime $\Gamma$-ring and $C_{\gamma}$ the extended centroid of M. If $C_{\gamma}$ is a $\Gamma$-field, then the $\Gamma$-ring M is a prime $\Gamma$-ring.