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Local Rings, Semilocal Rings, and Idempotents

open access: yesGraduate Texts in Mathematics, 1991
In the first two sections of this chapter, we focus our attention on two special classes of rings, namely, local rings and semilocal rings. By definition, a ring R is local if R/rad R is a division ring, and R is semilocal if R/rad R is a semisimple ring.
T Y Lam
exaly   +5 more sources

Factorizations of elements in local rings and semilocal rings of finite type

open access: yesJournal of Algebra and Its Applications, 2019
Factorizations of ring elements are described by finite chains of principal ideals. We use the description of cyclically presented modules over local rings to study factorization of elements in local rings.
Arroyo Paniagua María José   +2 more
openaire   +3 more sources
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Semilocal rings whose adjoint group is locally supersoluble

Archiv der Mathematik, 2010
Let \(R\) be an associative ring, not necessarily with an identity element. Let \(R^{ad}\) be the adjoint semigroup of \(R\) under the operation \(a\circ b=a+b+ab\) for all \(a,b\in R\) with neutral element \(0\in R\). Let \(R^\circ\) be the adjoint group of \(R\), that is, the group of all invertible elements of the semigroup \(R^{ad}\).
CATINO, Francesco   +2 more
openaire   +2 more sources

Characterizations of taut semi-local rings

Annali di Matematica Pura ed Applicata, 1977
It is proved that the following statements are equivalent for semi-local domain R:1) R is taut (i.e., for each non-maximal prime ideal P in R, height P+depth P=altitude R).2) Every integral domain which contains and is integral over R is taut.3) R[1/b].
openaire   +2 more sources

Projections of Semilocal Rings

Algebra and Logic, 2022
S S Korobkov
exaly  

Springer’s Odd Degree Extension Theorem for quadratic forms over semilocal rings

Indagationes Mathematicae, 2021
Erhard Neher, Philippe Gille
exaly  

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