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1.
《代数通讯》2013,41(12):4821-4833
Abstract In this note, we show that the following are equivalent for a ring R for which the socle or the injective hull of R R is finitely generated: (i) The direct sum of any two CS right R-modules is again CS; (ii) R is right Artinian and every uniform right R-module has composition length at most two. Next we give partial answers to a question of Huynh whether a right countably Σ-CS ring which either is semilocal or has finite Goldie dimension is right Σ-CS. We give characterizations, in terms of radicals, of when such rings are right Σ-CS. In particular, for the semilocal case, Huynh's question is reduced to whether rad(Z 2(R R )) is Σ-CS or Noetherian, where Z 2(R R ) is the second singular right ideal of R. Our results yield new characterizations of QF-rings. 相似文献
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《代数通讯》2013,41(10):5105-5116
Abstract A ring R is called left IP-injective if every homomorphism from a left ideal of R into R with principal image is given by right multiplication by an element of R. It is shown that R is left IP-injective if and only if R is left P-injective and left GIN (i.e., r(I ∩ K) = r(I) + r(K) for each pair of left ideals I and K of R with I principal). We prove that R is QF if and only if R is right noetherian and left IP-injective if and only if R is left perfect, left GIN and right simple-injective. We also show that, for a right CF left GIN-ring R, R is QF if and only if Soc(R R ) ? Soc( R R). Two examples are given to show that an IP-injective ring need not be self-injective and a right IP-injective ring is not necessarily left IP-injective respectively. 相似文献
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Miao-Sen Chen 《Southeast Asian Bulletin of Mathematics》2000,24(1):25-29
In this paper, we shall discuss the conditions for a right SC right CS ring to be a QF ring. In particular, we prove that if R is a right SI right CS ring satisfying the reflexive orthogonal condition (*) and if every CS right R-module is -CS, then R is a QF ring.AMS Subject Classification (1991): 16L30 16L60 相似文献
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Mathematical Notes - 相似文献
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It is well known that every serial Noetherian ring satisfies the restricted minimum condition. In particular, following Warfield (1975), such a ring is a direct sum of an Artinian ring and hereditary prime rings. The aim of this note is to show that every serial ring having the restricted minimum condition is Noetherian. 相似文献
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Dinh Van Huynh 《代数通讯》2013,41(3):984-987
Carl Faith in 2003 introduced and investigated an interesting class of rings over which every cyclic right module has Σ-injective injective hull (abbr., right CSI-rings) [5]. Inspired by this we investigate rings over which every cyclic right R-module has a projective Σ-injective injective hull. We show that a ring R satisfies this condition if and only if R is right artinian, the injective hull of R R is projective and every simple right R-module is embedded in R R . We also characterize right artinian rings in terms of injective faithful right ideals and right CSI-rings. 相似文献
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Wolfgang Rump 《Algebras and Representation Theory》2004,7(4):395-417
The method of differentiation for the category -lat of lattices over an order will be extended to integral almost Abelian categories A instead of -lat. In particular, this yields a differentiation for finitely generated left modules over left Artinian rings. 相似文献
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In this paper we introduce a construction called the skew generalized power series ring R[[S, ω]] with coefficients in a ring R and exponents in a strictly ordered monoid S which extends Ribenboim's construction of generalized power series rings. In the case when S is totally ordered or commutative aperiodic, and ω(s) is constant on idempotents for some s ∈ S?{1}, we give sufficient and necessary conditions on R and S such that the ring R[[S, ω]] is von Neumann regular, and we show that the von Neumann regularity of the ring R[[S, ω]] is equivalent to its semisimplicity. We also give a characterization of the strong regularity of the ring R[[S, ω]]. 相似文献
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本文证明了如下结果:环R是Artin半单的当且仅当存在一个基数c,使得任意左R-模是一个连续模和一个c-限制的ES-模的直和,也当且仅当存在一个基数c,使得任意左界R-模是一个拟投射模和一个c-限制的ES-模的直和。 相似文献
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Michiel Kosters 《代数通讯》2013,41(11):4911-4931
Let V be a finite-dimensional vector space over a field k, and let W be a 1-dimensional k-vector space. Let ?,?: V × V → W be a symmetric bilinear form. Then ?,? is called anisotropic if for all nonzero v ∈ V we have ? v, v ? ≠ 0. Motivated by a problem in algebraic number theory, we give a generalization of the concept of anisotropy to symmetric bilinear forms on finitely generated modules over artinian principal ideal rings. We will give many equivalent definitions of this concept of anisotropy. One of the definitions shows that a form is anisotropic if and only if certain forms on vector spaces are anisotropic. We will also discuss the concept of quasi-anisotropy of a symmetric bilinear form, which has no vector space analogue. Finally, we will discuss the radical root of a symmetric bilinear form, which does not have a vector space analogue either. All three concepts have applications in algebraic number theory. 相似文献
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周德旭 《数学物理学报(A辑)》2006,26(5):707-715
证明了环的有限扩张性可以传递到矩阵环上;通过PP环,半遗传环以及有限余非奇异环刻划了有限扩张环,并推广了文献[2]的定理2.1; 对于FGF与CF猜测,给出了部分肯定的回答,即右有限扩张右CF环是右CEP的,从而是右aritian的,改进了文献[6]的定理3.7. 相似文献
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Keisuke Shiromoto 《Journal of Algebraic Combinatorics》2000,12(1):95-99
We introduce the Singleton bounds for codes over a finite commutative quasi-Frobenius ring. 相似文献
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关于环的极大本质右理想 总被引:7,自引:0,他引:7
设R为环,我们考虑下面两个条件。(*)R的每个极大本质右理想是GP-内射右R-模或右零化子.(*)R的每个极大本质右理想是YJ-内射右R-模.本文旨在研究满足条件(*)或(*)的环,同时我们还给出了强正则环和除环的一些新刻画. 相似文献
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In this article we show, among others, that if R is a prime ring which is not a domain, then R is right nonsingular, right max-min CS with uniform right ideal if and only if R is left nonsingular, left max-min CS with uniform left ideal. The above result gives, in particular, Huynh et al. (2000) Theorem for prime rings of finite uniform dimension. 相似文献
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Tsiu-Kwen Lee 《代数通讯》2013,41(7):2923-2927
Let R be a semiprime ring with Q ml (R) the maximal left ring of quotients of R. Suppose that T: R → Q ml (R) is an additive map satisfying T(x 2) = xT(x) for all x ∈ R. Then T is a right centralizer; that is, there exists a ∈ Q ml (R) such that T(x) = xa for all x ∈ R. 相似文献