Prime ideals and radicals of centred extensions and tensor products
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.
scientific article; zbMATH DE number 902630
scientific article