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A ring is called clean if every element is the sum of an idempotent and a unit. It is shown that the endomorphism ring of a projective right module over a right perfect ring is clean.Received: 6 January 2003 相似文献
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K. M. Rangaswamy 《Mathematische Annalen》1970,190(2):167-176
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B. J. Gardner 《Acta Mathematica Hungarica》2009,124(4):371-383
We consider the determination of ring radicals by classes of modules as first discussed by Andrunakievich and Ryabukhin, but in cases where the modules concerned are defined by additive properties. Such a radical is the upper radical defined by the class of subrings of a class of endomorphism rings of abelian groups and is therefore strict. Not every strict radical is of this type, and while the A-radicals are of this type, there are others, including some special radicals. These investigations bring radical theory into contact with (at least) two questions from other parts of algebra. Which rings are endomorphism rings? For a given ring R, which abelian groups are non-trivial R-modules? 相似文献
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We study the endomorphism ringS of a Σ-quasiprojective moduleM, giving necessary and sufficient conditions onM forS to have certain properties, such as, e.g., being QF or left (F)PF. 相似文献
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Li-Xin Mao 《Proceedings Mathematical Sciences》2008,118(4):557-567
A ring is called right P-coherent if every principal right ideal is finitely presented. Let M
R
be a right R-module. We study the P-coherence of the endomorphism ring S of M
R
. It is shown that S is a right P-coherent ring if and only if every endomorphism of M
R
has a pseudokernel in add M
R
; S is a left P-coherent ring if and only if every endomorphism of M
R
has a pseudocokernel in add M
R
. Some applications are given. 相似文献
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Dr. Carlton J. Maxson 《Mathematische Zeitschrift》1971,122(4):294-298
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Eklof and Shelah [8] call an abelian group absolutely indecomposable if it is directly indecomposable in every generic extension of the universe. More generally, we say that an R-module is absolutely rigid if its endomorphism ring is just the ring of scalar multiplications by elements of R in every generic extension of the universe. In [8] it is proved that there do not exist absolutely rigid abelian groups of size ≥ κ(ω), where κ(ω) is the first ω-Erd?s cardinal (for its definition see the introduction). A similar result holds for rigid systems of abelian groups. On the other hand, recently Göbel and Shelah [15] proved that for modules of size < κ(ω) this phenomenon disappears. Their result on R ω -modules (i.e. on R-modules with countably many distinguished submodules) that establishes the existence of ‘well-behaving’ fully rigid systems of abelian groups of large sizes < κ(ω) will be extended here to a large class of R-modules by proving the existence of modules of any sizes < κ(ω) with endomorphism rings which are absolute. In order to cover rings as general as possible, we utilize a method developed by Brenner, Butler and Corner (see [2, 3, 5]) to reduce the number of distinguished submodules required in the construction from ?0 to five.We give several applications of our results. They include modules over domains with four pairwise comaximal prime elements, and modules over quasi-local rings whose completions contain at least five algebraically independent elements. 相似文献
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Patrik Lundström 《Indagationes Mathematicae》2003,14(2):223-232
Let K be a field of characteristic zero, G a group acting on a nonempty set X and KX the permutation module induced by this action. By studying traces of idempotents, we prove that the endomorphism ring EndK[G](KX) is von Neumann finite under certain conditions for the action of G on X. This generalizes a classical result by Kaplansky for the group ring of G over K. 相似文献
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Hiroshi Yoshimura 《代数通讯》2013,41(11):2663-2682
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Wu Tongsuo 《数学学报(英文版)》1998,14(3):385-390
In this paper, we study the endomorphism rings of regular modules. We give sufficient conditions on a regular projective moduleP such that EndR (P) has stable range one.
Dedicated to Professor Zhou Boxun for his 80'th Birthday
The author is supported by the NNSF of China (No. 19601009) 相似文献