Noetherian PI Rings not Module-Finite Over any Commutative Subring

1982 ◽  
Vol 84 (4) ◽  
pp. 457 ◽  
Author(s):  
J. J. Sarraille
Keyword(s):  
Commutative Subring

scholarly journals Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics

2019 ◽  
Vol 31 (3) ◽  
pp. 543-562 ◽  
Author(s):  
Viviane Beuter ◽  
Daniel Gonçalves ◽  
Johan Öinert ◽  
Danilo Royer
Keyword(s):  
Inverse Semigroup ◽  
Semigroup Ring ◽  
Topological Dynamics ◽  
Topological Properties ◽  
Semigroup Rings ◽  
Partial Action ◽  
Semigroup Actions ◽  
Formula One ◽  
Topological Inverse Semigroup ◽  
Commutative Subring

Abstract Given a partial action π of an inverse semigroup S on a ring {\mathcal{A}} , one may construct its associated skew inverse semigroup ring {\mathcal{A}\rtimes_{\pi}S} . Our main result asserts that, when {\mathcal{A}} is commutative, the ring {\mathcal{A}\rtimes_{\pi}S} is simple if, and only if, {\mathcal{A}} is a maximal commutative subring of {\mathcal{A}\rtimes_{\pi}S} and {\mathcal{A}} is S-simple. We apply this result in the context of topological inverse semigroup actions to connect simplicity of the associated skew inverse semigroup ring with topological properties of the action. Furthermore, we use our result to present a new proof of the simplicity criterion for a Steinberg algebra {A_{R}(\mathcal{G})} associated with a Hausdorff and ample groupoid {\mathcal{G}} .

scholarly journals The general Ikehata theorem forH-separable crossed products

2000 ◽  
Vol 23 (10) ◽  
pp. 657-662
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Automorphism Group ◽  
Galois Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Separable Extension ◽  
Factor Set ◽  
Commutative Subring

LetBbe a ring with1,   Cthe center ofB,   Gan automorphism group ofBof ordernfor some integern,   CGthe set of elements inCfixed underG,   Δ=Δ(B,G,f)a crossed product overBwherefis a factor set fromG×GtoU(CG). It is shown thatΔis anH-separable extension ofBandVΔ(B)is a commutative subring ofΔif and only ifCis a Galois algebra overCGwith Galois groupG|C≅G.

scholarly journals MAXIMAL COMMUTATIVE SUBRINGS AND SIMPLICITY OF ORE EXTENSIONS

2013 ◽  
Vol 12 (04) ◽  
pp. 1250192 ◽  
Author(s):  
JOHAN ÖINERT ◽  
JOHAN RICHTER ◽  
SERGEI D. SILVESTROV
Keyword(s):  
Polynomial Ring ◽  
Sufficient Conditions ◽  
Differential Polynomial ◽  
Nonzero Ideal ◽  
Polynomial Rings ◽  
Ore Extension ◽  
Differential Polynomial Ring ◽  
Ore Extensions ◽  
Necessary And Sufficient ◽  
Commutative Subring

The aim of this paper is to describe necessary and sufficient conditions for simplicity of Ore extension rings, with an emphasis on differential polynomial rings. We show that a differential polynomial ring, R[x; id R, δ], is simple if and only if its center is a field and R is δ-simple. When R is commutative we note that the centralizer of R in R[x; σ, δ] is a maximal commutative subring containing R and, in the case when σ = id R, we show that it intersects every nonzero ideal of R[x; id R, δ] nontrivially. Using this we show that if R is δ-simple and maximal commutative in R[x; id R, δ], then R[x; id R, δ] is simple. We also show that under some conditions on R the converse holds.

Simplicity of iterated Ore extensions

2021 ◽  
pp. 2250052
Author(s):  
Mamta Balodi ◽  
Sumit Kumar Upadhyay
Keyword(s):  
Polynomial Ring ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Differential Polynomial ◽  
Ore Extension ◽  
Differential Polynomial Ring ◽  
Ore Extensions ◽  
Commutative Domain ◽  
Necessary And Sufficient ◽  
Commutative Subring

Here we study the simplicity of an iterated Ore extension of a unital ring [Formula: see text]. We give necessary conditions for the simplicity of an iterated Ore extension when [Formula: see text] is a commutative domain. A class of iterated Ore extensions, namely the differential polynomial ring [Formula: see text] in [Formula: see text]-variables is considered. The conditions for a commutative domain [Formula: see text] of characteristic zero to be a maximal commutative subring of its differential polynomial ring [Formula: see text] are given, and the necessary and sufficient conditions for [Formula: see text] to be simple are also found.

On irresolute topological rings

2018 ◽  
Vol 9 (2) ◽  
pp. 130-134
Author(s):  
Haval Mahmood Mohammed Salih
Keyword(s):  
Hausdorff Space ◽  
Topological Ring ◽  
New Type ◽  
Commutative Subring

In this paper we introduce a new type of a topological ring which is an irresolute topological ring (semi topological ring). The relation among of them are studied. Several results are given. In particular, in a semi Hausdorff space, we show that if a subring is commutative, then its semi closure commutative subring. Furthermore, we show that the center of a ring is semi closed.

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