• Title/Summary/Keyword: Zero divisors

Search Result 38, Processing Time 0.018 seconds

ON v-MAROT MORI RINGS AND C-RINGS

  • Geroldinger, Alfred;Ramacher, Sebastian;Reinhart, Andreas
    • Journal of the Korean Mathematical Society
    • /
    • v.52 no.1
    • /
    • pp.1-21
    • /
    • 2015
  • C-domains are defined via class semigroups, and every C-domain is a Mori domain with nonzero conductor whose complete integral closure is a Krull domain with finite class group. In order to extend the concept of C-domains to rings with zero divisors, we study v-Marot rings as generalizations of ordinary Marot rings and investigate their theory of regular divisorial ideals. Based on this we establish a generalization of a result well-known for integral domains. Let R be a v-Marot Mori ring, $\hat{R}$ its complete integral closure, and suppose that the conductor f = (R : $\hat{R}$) is regular. If the residue class ring R/f and the class group C($\hat{R}$) are both finite, then R is a C-ring. Moreover, we study both v-Marot rings and C-rings under various ring extensions.

MATRIX RINGS AND ITS TOTAL RINGS OF FRACTIONS

  • Lee, Sang-Cheol
    • Honam Mathematical Journal
    • /
    • v.31 no.4
    • /
    • pp.515-527
    • /
    • 2009
  • Let R be a commutative ring with identity. Then we prove $M_n(R)=GL_n(R)$ ${\cup}${$A{\in}M_n(R)\;{\mid}\;detA{\neq}0$ and det $A{\neq}U(R)$}${\cup}Z(M-n(R))$ where U(R) denotes the set of all units of R. In particular, it will be proved that the full matrix ring $M_n(F)$ over a field F is the disjoint union of the general linear group $GL_n(F)$ of degree n over the field F and the set $Z(M_n(F))$ of all zero-divisors of $M_n(F)$. Using the result and universal mapping property we prove that $M_n(F)$ is its total ring of fractions.

ANNIHILATING PROPERTY OF ZERO-DIVISORS

  • Jung, Da Woon;Lee, Chang Ik;Lee, Yang;Nam, Sang Bok;Ryu, Sung Ju;Sung, Hyo Jin;Yun, Sang Jo
    • Communications of the Korean Mathematical Society
    • /
    • v.36 no.1
    • /
    • pp.27-39
    • /
    • 2021
  • We discuss the condition that every nonzero right annihilator of an element contains a nonzero ideal, as a generalization of the insertion-of-factors-property. A ring with such condition is called right AP. We prove that a ring R is right AP if and only if Dn(R) is right AP for every n ≥ 2, where Dn(R) is the ring of n by n upper triangular matrices over R whose diagonals are equal. Properties of right AP rings are investigated in relation to nilradicals, prime factor rings and minimal order.

ERRATUM TO "RINGS IN WHICH EVERY IDEAL CONTAINED IN THE SET OF ZERO-DIVISORS IS A D-IDEAL", COMMUN. KOREAN MATH. SOC. 37 (2022), NO. 1, PP. 45-56

  • Adam Anebri;Najib Mahdou;Abdeslam Mimouni
    • Communications of the Korean Mathematical Society
    • /
    • v.38 no.1
    • /
    • pp.121-122
    • /
    • 2023
  • In this erratum, we correct a mistake in the proof of Proposition 2.7. In fact the equivalence (3) ⇐ (4) "R is a quasi-regular ring if and only if R is a reduced ring and every principal ideal contained in Z(R) is a 0-ideal" does not hold as we only have Rx ⊆ O(S).

THE CHOW RING OF A SEQUENCE OF POINT BLOW-UPS

  • Daniel Camazon Portela
    • Communications of the Korean Mathematical Society
    • /
    • v.39 no.3
    • /
    • pp.563-574
    • /
    • 2024
  • Given a sequence of point blow-ups of smooth n-dimensional projective varieties Zi defined over an algebraically closed field $k,\,Z_s\,{\overset{{\pi}_s}{\longrightarrow}}\,Z_{s-1}\,{\overset{{\pi}_{s-1}}{\longrightarrow}}\,{\cdots}\,{\overset{{\pi}_2}{\longrightarrow}}\,Z_1\,{\overset{{\pi}_1}{\longrightarrow}}\,Z_0$, with Z0 ≅ ℙn, we give two presentations of the Chow ring A(Zs) of its sky. The first one uses the classes of the total transforms of the exceptional components as generators and the second one uses the classes of the strict transforms ones. We prove that the skies of two sequences of point blow-ups of the same length have isomorphic Chow rings. Finally we give a characterization of the final divisors of a sequence of point blow-ups in terms of some relations defined over the Chow group of zero-cycles A0(Zs) of its sky.

GRADED PSEUDO-VALUATION RINGS

  • Fatima-Zahra Guissi;Hwankoo Kim;Najib Mahdou
    • Journal of the Korean Mathematical Society
    • /
    • v.61 no.5
    • /
    • pp.953-973
    • /
    • 2024
  • Let R = ⊕α∈Γ Rα be a commutative ring graded by an arbitrary torsionless monoid Γ. A homogeneous prime ideal P of R is said to be strongly homogeneous prime if aP and bR are comparable for any homogeneous elements a, b of R. We will say that R is a graded pseudo-valuation ring (gr-PVR for short) if every homogeneous prime ideal of R is strongly homogeneous prime. In this paper, we introduce and study the graded version of the pseudo-valuation rings which is a generalization of the gr-pseudo-valuation domains in the context of arbitrary Γ-graded rings (with zero-divisors). We then study the possible transfer of this property to the graded trivial ring extension and the graded amalgamation. Our goal is to provide examples of new classes of Γ-graded rings that satisfy the above mentioned property.

An Investigation on the Historical Developments of the Algorithms for Multiplication of Natural Numbers (자연수 곱셈 계산법의 역사적 발달 과정에 대한 고찰)

  • Joung, Youn-Joon
    • School Mathematics
    • /
    • v.13 no.2
    • /
    • pp.267-286
    • /
    • 2011
  • In this paper I investigated the historical developments of the algorithms for multiplication of natural numbers. Through this analysis I tried to describe more concretely what is to understand the common algorithm for multiplication of natural numbers. I found that decomposing dividends and divisors into small numbers and multiplying these numbers is the main strategy for carrying out multiplication of large numbers, and two decomposing and multiplying processes are very important in the algorithms for multiplication. Finally I proposed some implications based on these analysis.

  • PDF

ON 𝜙-SCHREIER RINGS

  • Darani, Ahmad Yousefian;Rahmatinia, Mahdi
    • Journal of the Korean Mathematical Society
    • /
    • v.53 no.5
    • /
    • pp.1057-1075
    • /
    • 2016
  • Let R be a ring in which Nil(R) is a divided prime ideal of R. Then, for a suitable property X of integral domains, we can define a ${\phi}$-X-ring if R/Nil(R) is an X-domain. This device was introduced by Badawi [8] to study rings with zero divisors with a homomorphic image a particular type of domain. We use it to introduce and study a number of concepts such as ${\phi}$-Schreier rings, ${\phi}$-quasi-Schreier rings, ${\phi}$-almost-rings, ${\phi}$-almost-quasi-Schreier rings, ${\phi}$-GCD rings, ${\phi}$-generalized GCD rings and ${\phi}$-almost GCD rings as rings R with Nil(R) a divided prime ideal of R such that R/Nil(R) is a Schreier domain, quasi-Schreier domain, almost domain, almost-quasi-Schreier domain, GCD domain, generalized GCD domain and almost GCD domain, respectively. We study some generalizations of these concepts, in light of generalizations of these concepts in the domain case, as well. Here a domain D is pre-Schreier if for all $x,y,z{\in}D{\backslash}0$, x | yz in D implies that x = rs where r | y and s | z. An integrally closed pre-Schreier domain was initially called a Schreier domain by Cohn in [15] where it was shown that a GCD domain is a Schreier domain.