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陈焕艮 《数学物理学报(A辑)》2014,(6)
设R是一个环,J(R)表示R的Jacbson根.R的一个元素称为强J-clean的,如果能够表示成一个幂等元和一个J(R)中元素的和且这两个元素可交换.对于一个可交换局部环R满足2∈J(R),得到一个在RG上2×2矩阵是强J-clean的充要条件,其中G={1,g}是一个群.同时给出了强clean性的上应用. 相似文献
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设R为有1结合环.G为群,H为G的子群,本文对群环RG的Jacobson根作了一些刻化,从而得出了在不同条件下J(RG)、J(RH)、(RG)之间的一些关系式:J(RG)=J(RH)RG.J(RH)RG=(RG),J(RH)=(RH)=UJ(RW)等,其中J(RG)、(RG)分别表示RG的Jacobson根和Baer根. 相似文献
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Kegel曾经提出如下猜测:若环R可以表示为它的两个局部幂零子环S、T之和,即有R=S+T,问R是否必是局部幂零的?本文证明:若Kegel猜测不真,则必存在一个本原环可以表示为它的两个局部幂零子环之和.另外,还得到两个与Kegel猜测有关的很有趣的结果. 相似文献
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交换环R称为(受限制的)弱准素环,如果R中的每个(非零)主理想都是准素理想.本文证明了一个没有单位元的交换环R是受限制的弱准素环当且仅当R是每个元素都是幂零元的交换环或者R是仅含一个真素理想P的没有单位元的交换环并且P不真包含R的任何非零理想. 相似文献
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AbstractRings in which the square of each unit is a sum of an idempotent and an element from the Jacobson radical are said to be 2-UJ. Properties of 2-UJ rings are discussed, and the 2-UJ property is applied to characterize some known notions of rings. 相似文献
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Motivated by Hirano-Tominaga’s work on rings for which every element is a sum of two idempotents and by de Seguins Pazzis’s results on decomposing every matrix over a field of positive characteristic as a sum of idempotent matrices, we address decomposing every matrix over a commutative ring as a sum of three idempotent matrices and, respectively, as a sum of three involutive matrices. 相似文献
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《代数通讯》2013,41(7):3295-3304
Abstract An element in a ring is called clean if it may be written as a sum of a unit and idempotent. The ring itself is called clean if every element is clean. Recently,Anderson and Camillo (Anderson,D. D.,Camillo,V. (2002). Commutative rings whose elements are a sum of a unit and an idempotent. Comm. Algebra 30(7):3327–3336) has shown that for commutative rings every von-Neumann regular ring as well as zero-dimensional rings are clean. Moreover,every clean ring is a pm-ring,that is every prime ideal is contained in a unique maximal ideal. In the same article,the authors give an example of a commutative ring which is a pm-ring yet not clean,e.g.,C(?). It is this example which interests us. Our discussion shall take place in a more general setting. We assume that all rings are commutative with 1. 相似文献
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Huanyin Chen 《Czechoslovak Mathematical Journal》2008,58(4):899-910
Let R be an exchange ring in which all regular elements are one-sided unit-regular. Then every regular element in R is the sum of an idempotent and a one-sided unit. Furthermore, we extend this result to exchange rings satisfying related
comparability. 相似文献
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This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings.The properties of radicals of pseudo-reduced-over-center rings are investigated,especially related to polynomial rings.It is proved that for pseudo-reduced-over-center rings of nonzero characteristic,the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals.For a locally finite ring R,it is proved that if R is pseudo-reduced-over-center,then R is commutative and R/J(R) is a commutative regular ring with J(R) nil,where J(R) is the Jacobson radical of R. 相似文献
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陆仲坚 《纯粹数学与应用数学》2001,17(2):175-179
首先给出了 gr-正则环为分次除环的两个充要条件 ,其次讨论了分次正则环r G(R)和分次 Jacobson根 JG(R)之间的关系 ,最后给出了分次 Abel正则环的结构定理 . 相似文献
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A *-ring is called *-clean if every element of the ring can be written as the sum of a projection and a unit. For an integer n ≥ 1, we call a *-ring R n-*-clean if for any a ∈ R,a = p + u1 + ··· + unwhere p is a projection and ui are units for all i. Basic properties of n-*-clean rings are considered, and a number of illustrative examples of 2-*-clean rings which are not *-clean are provided. In addition, extension properties of n-*-clean rings are discussed. 相似文献
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Lingling Fan 《代数通讯》2013,41(1):269-278
A ring R with identity is called “clean” if for every element a ? R there exist an idempotent e and a unit u in R such that a = e + u. Let C(R) denote the center of a ring R and g(x) be a polynomial in the polynomial ring C(R)[x]. An element r ? R is called “g(x)-clean” if r = s + u where g(s) = 0 and u is a unit of R and R is g(x)-clean if every element is g(x)-clean. Clean rings are g(x)-clean where g(x) ? (x ? a)(x ? b)C(R)[x] with a, b ? C(R) and b ? a ? U(R); equivalent conditions for (x2 ? 2x)-clean rings are obtained; and some properties of g(x)-clean rings are given. 相似文献
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A ring R is a Garcia ring provided that the product of two regular elements is unit-regular. We prove that every regular element in a Garcia ring R is the sum/difference of an idempotent and a unit. Furthermore, we prove that every regular element in a weak Garcia ring is the sum of an idempotent and a one-sided unit. These extend several known theorems on (one-sided) unit-regular rings to wider classes of rings with sum summand property. 相似文献
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Javier Majadas 《代数通讯》2013,41(2):737-746
It is shown that every element in a one-sided unit-regular ring is the sum of an idempotent and a one-sided unit. Also we prove that every element in a regular ring satisfying related comparability is the sum of an idempotent and the product of two one-sided units. 相似文献