• 제목/요약/키워드: prime factor ring

검색결과 14건 처리시간 0.022초

Commutativity Criteria for a Factor Ring R/P Arising from P-Centralizers

  • Lahcen Oukhtite;Karim Bouchannafa;My Abdallah Idrissi
    • Kyungpook Mathematical Journal
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    • 제63권4호
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    • pp.551-560
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    • 2023
  • In this paper we consider a more general class of centralizers called I-centralizers. More precisely, given a prime ideal P of an arbitrary ring R we establish a connection between certain algebraic identities involving a pair of P-left centralizers and the structure of the factor ring R/P.

Structures Related to Right Duo Factor Rings

  • Chen, Hongying;Lee, Yang;Piao, Zhelin
    • Kyungpook Mathematical Journal
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    • 제61권1호
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    • pp.11-21
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    • 2021
  • We study the structure of rings whose factor rings modulo nonzero proper ideals are right duo; such rings are called right FD. We first see that this new ring property is not left-right symmetric. We prove for a non-prime right FD ring R that R is a subdirect product of subdirectly irreducible right FD rings; and that R/N∗(R) is a subdirect product of right duo domains, and R/J(R) is a subdirect product of division rings, where N∗(R) (J(R)) is the prime (Jacobson) radical of R. We study the relation among right FD rings, division rings, commutative rings, right duo rings and simple rings, in relation to matrix rings, polynomial rings and direct products. We prove that if a ring R is right FD and 0 ≠ e2 = e ∈ R then eRe is also right FD, examining that the class of right FD rings is not closed under subrings.

ON RIGHT REGULARITY OF COMMUTATORS

  • Jung, Da Woon;Lee, Chang Ik;Lee, Yang;Park, Sangwon;Ryu, Sung Ju;Sung, Hyo Jin
    • 대한수학회보
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    • 제59권4호
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    • pp.853-868
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    • 2022
  • We study the structure of right regular commutators, and call a ring R strongly C-regular if ab - ba ∈ (ab - ba)2R for any a, b ∈ R. We first prove that a noncommutative strongly C-regular domain is a division algebra generated by all commutators; and that a ring (possibly without identity) is strongly C-regular if and only if it is Abelian C-regular (from which we infer that strong C-regularity is left-right symmetric). It is proved that for a strongly C-regular ring R, (i) if R/W(R) is commutative, then R is commutative; and (ii) every prime factor ring of R is either a commutative domain or a noncommutative division ring, where W(R) is the Wedderburn radical of R.

The Factor Domains that Result from Uppers to Prime Ideals in Polynomial Rings

  • Dobbs, David Earl
    • Kyungpook Mathematical Journal
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    • 제50권1호
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    • pp.1-5
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    • 2010
  • Let P be a prime ideal of a commutative unital ring R; X an indeterminate; D := R/P; L the quotient field of D; F an algebraic closure of L; ${\alpha}$ ${\in}$ L[X] a monic irreducible polynomial; ${\xi}$ any root of in F; and Q = ${\alpha}$>, the upper to P with respect to ${\alpha}$. Then R[X]/Q is R-algebra isomorphic to $D[{\xi}]$; and is R-isomorphic to an overring of D if and only if deg(${\alpha}$) = 1.

RELATIONSHIP BETWEEN THE STRUCTURE OF A FACTOR RING R/P AND DERIVATIONS OF R

  • Karim Bouchannafa;Moulay Abdallah Idrissi;Lahcen Oukhtite
    • 대한수학회보
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    • 제60권5호
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    • pp.1281-1293
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    • 2023
  • The purpose of this paper is to study the relationship between the structure of a factor ring R/P and the behavior of some derivations of R. More precisely, we establish a connection between the commutativity of R/P and derivations of R satisfying specific identities involving the prime ideal P. Moreover, we provide an example to show that our results cannot be extended to semi-prime ideals.

ARMENDARIZ PROPERTY OVER PRIME RADICALS

  • Han, Juncheol;Kim, Hong Kee;Lee, Yang
    • 대한수학회지
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    • 제50권5호
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    • pp.973-989
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    • 2013
  • We observe from known results that the set of nilpotent elements in Armendariz rings has an important role. The upper nilradical coincides with the prime radical in Armendariz rings. So it can be shown that the factor ring of an Armendariz ring over its prime radical is also Armendariz, with the help of Antoine's results for nil-Armendariz rings. We study the structure of rings with such property in Armendariz rings and introduce APR as a generalization. It is shown that APR is placed between Armendariz and nil-Armendariz. It is shown that an APR ring which is not Armendariz, can always be constructed from any Armendariz ring. It is also proved that a ring R is APR if and only if so is R[$x$], and that N(R[$x$]) = N(R)[$x$] when R is APR, where R[$x$] is the polynomial ring with an indeterminate $x$ over R and N(-) denotes the set of all nilpotent elements. Several kinds of APR rings are found or constructed in the precess related to ordinary ring constructions.

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
    • 대한수학회논문집
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    • 제36권1호
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    • pp.27-39
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    • 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.

ON WEAK II-REGULARITY AND THE SIMPLICITY OF PRIME FACTOR RINGS

  • Kim, Jin-Yong;Jin, Hai-Lan
    • 대한수학회보
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    • 제44권1호
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    • pp.151-156
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    • 2007
  • A connection between weak ${\pi}-regularity$ and the condition every prime ideal is maximal will be investigated. We prove that a certain 2-primal ring R is weakly ${\pi}-regular$ if and only if every prime ideal is maximal. This result extends several known results nontrivially. Moreover a characterization of minimal prime ideals is also considered.

RINGS WITH A RIGHT DUO FACTOR RING BY AN IDEAL CONTAINED IN THE CENTER

  • Cheon, Jeoung Soo;Kwak, Tai Keun;Lee, Yang;Piao, Zhelin;Yun, Sang Jo
    • 대한수학회보
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    • 제59권3호
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    • pp.529-545
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    • 2022
  • This article concerns a ring property that arises from combining one-sided duo factor rings and centers. A ring R is called right CIFD if R/I is right duo by some proper ideal I of R such that I is contained in the center of R. We first see that this property is seated between right duo and right π-duo, and not left-right symmetric. We prove, for a right CIFD ring R, that W(R) coincides with the set of all nilpotent elements of R; that R/P is a right duo domain for every minimal prime ideal P of R; that R/W(R) is strongly right bounded; and that every prime ideal of R is maximal if and only if R/W(R) is strongly regular, where W(R) is the Wedderburn radical of R. It is also proved that a ring R is commutative if and only if D3(R) is right CIFD, where D3(R) is the ring of 3 by 3 upper triangular matrices over R whose diagonals are equal. Furthermore, we show that the right CIFD property does not pass to polynomial rings, and that the polynomial ring over a ring R is right CIFD if and only if R/I is commutative by a proper ideal I of R contained in the center of R.

ON A RING PROPERTY RELATED TO NILRADICALS

  • Jin, Hai-lan;Piao, Zhelin;Yun, Sang Jo
    • Korean Journal of Mathematics
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    • 제27권1호
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    • pp.141-150
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    • 2019
  • In this article we investigate the structure of rings in which lower nilradicals coincide with upper nilradicals. Such rings shall be said to be quasi-2-primal. It is shown first that the $K{\ddot{o}}the^{\prime}s$ conjecture holds for quasi-2-primal rings. So the results in this article may provide interesting and useful information to the study of nilradicals in various situations. In the procedure we study the structure of quasi-2-primal rings, and observe various kinds of quasi-2-primal rings which do roles in ring theory.