Prime ideals and radicals of centred extensions and tensor products
type: Item
instance of: scholarly article
title: Prime ideals and radicals of centred extensions and tensor products
zbMATH Open document ID: 0856.16022
DOI: 10.1007/BF02762713
author: Miguel Ferrero
author: E. R. Puczyłowski
published in: Israel Journal of Mathematics
publication date: 1997-02-12T00:00:00Z
review text: A ring extension \(R\subset S\) is called centered if \(S\) is generated, as module over \(R\), by some subset of \(C_S(R)=\{s\in S\mid sr=rs\) for all \(r\in R\}\). In case \(R\) is prime, with Martindale ring of quotients \(Q\), there is a canonical \(S\)-torsion-free extension \(S^*\supseteq\langle Q,S\rangle\) such that the set of \(R\)-disjoint prime ideals of \(S\) is in one-to-one correspondence with the set of all primes of \(V=C_{S^*}(Q)\) [\textit{M. Ferrero}, J. Algebra 172, No. 2, 470-505 (1995; Zbl 0826.16021)]. The article under review studies the relations between \(\alpha(S)\) and \(\alpha(V)\) for various radicals, including the prime, locally nilpotent, strongly prime, and Jacobson radicals. These results are then applied to investigate prime ideals in tensor products \(A\otimes_FB\) of algebras \(A\), \(B\) over a commutative ring \(F\). In particular, the following result is established (assuming without loss, that \(A\to A\otimes_FB\) and \(B\to A\otimes_FB\) are injective): \(A\otimes_FB\) is prime if and only if both \(A\) and \(B\) are prime, \(A\otimes_FB\) is torsion-free as \(A\)-module and as \(B\)-module, and \(C(A)\otimes_FC(B)\) is a domain. Here, \(C(\cdot)\) denotes Martindale's extended centroid.
Mathematics Subject Classification ID: 16S20
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16N80
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16S90
zbMATH DE Number: 902630
zbMATH Keywords: ring extensions
zbMATH Keywords: Martindale rings of quotients
zbMATH Keywords: prime ideals
zbMATH Keywords: radicals
zbMATH Keywords: tensor products
zbMATH Keywords: extended centroids
reviewed by: Martin Lorenz
MaRDI profile type: Publication
cites work: Q3828152
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description: scientific article; zbMATH DE number 902630