Spectrum of a Noncommutative Ring

Abstract

R is any ring with identity. Let Spec r R resp. Spec R be the set of all prime right ideals resp. all prime ideals of R and let U r eR = P ∈ Spec r R e P . In this article, we study the relationships among various ring-theoretic properties and topological conditions on Spec r R with weak Zariski topology . A ring R is called Abelian if all idempotents in R are central see Goodearl, 1991 . A ring R is called 2-primal if every nilpotent element is in the prime radical of R see Lam, 2001 . It will be shown that for an Abelian ring R there is a bijection between the set of all idempotents in R and the clopen i.e., closed and open sets in Spec r R . And the following results are obtained for any ring R: 1 For any clopen set U in Spec r R , there is an idempotent e in R such that U = U r eR . 2 If R is an Abelian ring or a 2-primal ring, then, for any idempotent e in R, U r eR is a clopen set in Spec r R . 3 Spec r R is connected if and only if Spec R is connected.

Clean elements in abelian rings

“…Let Spec r (R) be the set of all prime right ideals of R. It has been shown in Corollary 2.8 of [5], that if R is not a right quasi-duo ring, then Spec r (R) is a space with the weak Zariski topology but not with the Zariski topology. The weak Zariski topology on Spec r (R) has also been studied in [4,6]. Then τ contains the empty set and Spec r (R).…”

“…Remark. In Theorem 3.5 of [6], it has been shown that the set {U r (e)|e ∈ Id(R)} consists of all the clopen sets in Spec r (R).…”

On the prime spectrum of an le-module

“…In this article we introduce and study Zariski topology on the set Spec(M) of all prime submodule elements of an le-module R M. It is well established that the Zariski topology on prime spectrum is a very efficient tool to give geometric interpretation of the arithmetic in rings [2], [14], [25], [29], [36] and modules [1], [5], [6], [12], [13], [22], [23], [24], [26]. Here we have extended several results on Zariski topology in modules to le-modules.…”

Prime spectrum and primitive Leavitt path algebras

“…Throughout the mathematical literature, knowing the prime and primitive spectra of rings (also of associative, Lie and Jordan algebras, etc) has been crucial in order to succeed to give structural theorems (or in order to simply gain a better understanding of the given algebraic system). Classically, one of the uses of the prime spectrum for commutative rings is to carry information over from Algebra to Topology and vice versa via the so-called Zariski topology (several generalizations of this construction for noncommutative rings have been achieved [17,24]). As for the primitive ideals of a ring, they naturally correspond to the irreducible representations of it, which in turn represent unquestionable tools in their analysis.…”