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- 검색키워드
- 주제어 : matrix ring (검색결과 115 건)
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ON A GENERALIZATION OF THE MCCOY CONDITION
Jeon, Young-Cheol,Kim, Hong-Kee,Kim, Nam-Kyun,Kwak, Tai-Keun,Lee, Yang,Yeo, Dong-Eun Korean Mathematical Society 2010 대한수학회지 Vol.47 No.6
We in this note consider a new concept, so called $\pi$-McCoy, which unifies McCoy rings and IFP rings. The classes of McCoy rings and IFP rings do not contain full matrix rings and upper (lower) triangular matrix rings, but the class of $\pi$-McCoy rings contain upper (lower) triangular matrix rings and many kinds of full matrix rings. We first study the basic structure of $\pi$-McCoy rings, observing the relations among $\pi$-McCoy rings, Abelian rings, 2-primal rings, directly finite rings, and ($\pi-$)regular rings. It is proved that the n by n full matrix rings ($n\geq2$) over reduced rings are not $\pi$-McCoy, finding $\pi$-McCoy matrix rings over non-reduced rings. It is shown that the $\pi$-McCoyness is preserved by polynomial rings (when they are of bounded index of nilpotency) and classical quotient rings. Several kinds of extensions of $\pi$-McCoy rings are also examined.
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ON A GENERALIZATION OF THE MCCOY CONDITION
전영철,김홍기,김남균,곽태근,이양,여동은 대한수학회 2010 대한수학회지 Vol.47 No.6
We in this note consider a new concept, so called π-McCoy,which unifies McCoy rings and IFP rings. The classes of McCoy rings and IFP rings do not contain full matrix rings and upper (lower) triangular matrix rings, but the class of π-McCoy rings contain upper (lower) triangular matrix rings and many kinds of full matrix rings. We first study the basic structure of π-McCoy rings, observing the relations among π-McCoy rings, Abelian rings, 2-primal rings, directly finite rings, and (π-)regular rings. It is proved that the n by n full matrix rings (n ¸ 2) over reduced rings are not π-McCoy, finding π-McCoy matrix rings over non-reduced rings. It is shown that the π-McCoyness is preserved by polynomial rings (when they are of bounded index of nilpotency) and classical quotient rings. Several kinds of extensions of π-McCoy rings are also examined.
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Matrix Rings over Reflexive Rings
Cheon, Jeoung Soo,Kwak, Tai Keun,Lee, Yang World Scientific Publishing Company 2018 Algebra Colloquium Vol.25 No.3
<P>The concept of reflexive property is introduced by Mason. This note concerns a ring-theoretic property of matrix rings over reflexive rings. We introduce the concept of <I>weakly reflexive</I> rings as a generalization of reflexive rings. From any ring, we can construct weakly reflexive rings but not reflexive, using its lower nilradical. We study various useful properties of such rings in relation with ideals in matrix rings, showing that this new property is Morita invariant. We also investigate the weakly reflexive property of several sorts of ring extensions which have roles in ring theory.</P>
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Rings with Property (A) and their extensions
Hong, C.Y.,Kim, N.K.,Lee, Y.,Ryu, S.J. Academic Press 2007 Journal of algebra Vol.315 No.2
A commutative ring R has Property (A) if every finitely generated ideal of R consisting entirely of zero-divisors has a nonzero annihilator. We continue in this paper the study of rings with Property (A). We extend Property (A) to noncommutative rings, and study such rings. Moreover, we study several extensions of rings with Property (A) including matrix rings, polynomial rings, power series rings and classical quotient rings. Finally, we characterize when the space of minimal prime ideals of rings with Property (A) is compact.
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f-CLEAN RINGS AND RINGS HAVING MANY FULL ELEMENTS
Bingjun Li,Lianggui Feng 대한수학회 2010 대한수학회지 Vol.47 No.2
An associative ring R with identity is called a clean ring if every element of R is the sum of a unit and an idempotent. In this paper,we introduce the concept of f-clean rings. We study various properties of f-clean rings. [수식]be a Morita Context ring. We determine conditions under which the ring C is f-clean. Moreover, we introduce the concept of rings having many full elements. We investigate characterizations of this kind of rings and show that rings having many full elements are closed under matrix rings and Morita Context rings.
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QUASIPOLAR MATRIX RINGS OVER LOCAL RINGS
Cui, Jian,Yin, Xiaobin Korean Mathematical Society 2014 대한수학회보 Vol.51 No.3
A ring R is called quasipolar if for every a 2 R there exists $p^2=p{\in}R$ such that $p{\in}comm^2{_R}(a)$, $ a+p{\in}U(R)$ and $ap{\in}R^{qnil}$. The class of quasipolar rings lies properly between the class of strongly ${\pi}$-regular rings and the class of strongly clean rings. In this paper, we determine when a $2{\times}2$ matrix over a local ring is quasipolar. Necessary and sufficient conditions for a $2{\times}2$ matrix ring to be quasipolar are obtained.
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f-CLEAN RINGS AND RINGS HAVING MANY FULL ELEMENTS
Li, Bingjun,Feng, Lianggui Korean Mathematical Society 2010 대한수학회지 Vol.47 No.2
An associative ring R with identity is called a clean ring if every element of R is the sum of a unit and an idempotent. In this paper, we introduce the concept of f-clean rings. We study various properties of f-clean rings. Let C = $\(\array{A\;V\\W\;B}\)$ be a Morita Context ring. We determine conditions under which the ring C is f-clean. Moreover, we introduce the concept of rings having many full elements. We investigate characterizations of this kind of rings and show that rings having many full elements are closed under matrix rings and Morita Context rings.
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Quasipolar matrix rings over local rings
Jian Cui,Xiaobin Yin 대한수학회 2014 대한수학회보 Vol.51 No.3
A ring R is called quasipolar if for every a 2 R there exists p2 = p ∈ R such that p ∈ comm2 R(a), a + p ∈ U(R) and ap 2∈Rqnil. The class of quasipolar rings lies properly between the class of strongly π-regular rings and the class of strongly clean rings. In this paper, we determine when a 2 × 2 matrix over a local ring is quasipolar. Necessary and sufficient conditions for a 2 × 2 matrix ring to be quasipolar are obtained.
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ON COMMUTATIVITY OF NILPOTENT ELEMENTS AT ZERO
Abdul-Jabbar, Abdullah M.,Ahmed, Chenar Abdul Kareem,Kwak, Tai Keun,Lee, Yang Korean Mathematical Society 2017 대한수학회논문집 Vol.32 No.4
The reversible property of rings was initially introduced by Habeb and plays a role in noncommutative ring theory. In this note we study the reversible ring property on nilpotent elements, introducing the concept of commutativity of nilpotent elements at zero (simply, a CNZ ring) as a generalization of reversible rings. We first find the CNZ property of 2 by 2 full matrix rings over fields, which provides a basis for studying the structure of CNZ rings. We next observe various kinds of CNZ rings including ordinary ring extensions.
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Zhelin Piao,류성주,성효진,윤상조 강원경기수학회 2020 한국수학논문집 Vol.28 No.1
This article concerns a property of local rings and domains. A ring $R$ is called {\it weakly local} if for every $a\in R$, $a$ is regular or $1-a$ is regular, where a regular element means a non-zero-divisor. We study the structure of weakly local rings in relation to several kinds of factor rings and ring extensions that play roles in ring theory. We prove that the characteristic of a weakly local ring is either zero or a power of a prime number. It is also shown that the weakly local property can go up to polynomial (power series) rings and a kind of Abelian matrix rings.
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