P −REGULAR AND P −LOCAL RINGS (original) (raw)
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The aim of this paper is to characterize those elements in a semiprime ring R for which taking local rings at elements and rings of quotients are commuting operations. If Q denotes the maximal ring of left quotients of R, then this happens precisely for those elements if R which are von Neumann regular in Q. An intrinsic characterization of such elements is given. We derive as a consequence that the maximal left quotient ring of a prime ring with a nonzero PI-element is primitive and has nonzero socle. If we change Q to the Martindale symmetric ring of quotients, or to the maximal symmetric ring of quotients of R, we obtain similar results: an element a in R is von Neumann regular if and only if the ring of quotients of the local ring of R at a is isomorphic to the local ring of Q at a.
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For a ring R and a right R-module M , a submodule N of M is said to be -small in M if, whenever N + X = M with M = Xsingular, we h a ve X = M . If there exists an epimorphism p : P ! M such that P is projective and Kerp i s -small in P , then we s a y that P is a projective -cover of M . A ring R is called -perfect resp., -semiperfect, -semiregular if every R-module resp., simple R-module, cyclically presented R-module has a projective -cover. The class of all -perfect resp., -semiperfect, -semiregular rings contains properly the class of all right perfect resp., semiperfect, semiregular rings. This paper is devoted to various properties and characterizations of -perfect, -semiperfect, and -semiregular rings. We de ne R by R=SocRR = JacR=SocRR and show, among others, the following results: 1 R is the largest -small right ideal of R.
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A ring R is called semiregular if R/J is regular and idempotents lift modulo J, where J denotes the Jacobson radical of R. We give some characterizations of rings R such that idempotents lift modulo J, and R/J satisfies one of the following conditions: (one-sided) unitregular, strongly regular, (unit, strongly, weakly) π-regular.
Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2016
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Pacific Journal of Mathematics, 1977
Several new properties are derived for von Neumann finite rings. A comparison is made of the properties of von Neumann finite regular rings and unit regular rings, and necessary and sufficient conditions are given for a matrix ring over a regular ring to be respectively von Neumann finite or unit regular. The converse of a theorem of Henriksen is proven, namely that if R n x n , the n x n matrix ring over ring R, is unit regular, then so is the ring R. It is shown that if R 2 2 is finite regular then a e R is unit regular if and only if there is x e R such that R -aRΛ-x(a°), where a 0 denotes the right annihilator of a in R.