Integrality for PI-rings
A ring \(S\) is called integral over a subring \(R\) if each element \(x\) of \(S\) satisfies an equation of the form \(x^ n+\sum p_ i(x)\), where the \(p_ i\) are words formed with \(x\) and elements of \(R\) with \(\text{deg}_ x(p_ i)<n\). This notion, introduced by \textit{W. Schelter} [ibid. 40, 245-257 (1976; Zbl 0341.16009)] for PI-rings, is thoroughly analyzed in the present paper. First, it is shown that lying over, going up, and incomparability hold for integral extensions of PI-rings, proving, among other things, that \(\text{Kdim}(R)= \text{Kdim}(S)\) if one of the two rings has classical Krull dimension \(\text{Kdim}\). Transitivity of integrality is another major result, generalizing earlier work by \textit{R. Paré} and \textit{W. Schelter} [ibid. 53, 477-479 (1978; Zbl 0404.16011)] for integral centralizing extensions, and by \textit{M. Artin} and \textit{W. Schelter} [Adv. Math. 39, 289-329 (1981; Zbl 0461.16014)] for affine PI-algebras over an algebraically closed field. A further topic are results of Eakin-Nagata type. Thus, it is shown that if \(R \subseteq S\) are PI-algebras over a commutative noetherian ring \(\Lambda\), such that \(S\) is left noetherian and affine over \(\Lambda\) and integral over \(R\), then \(_ RS\) is finitely generated and \(R\) is left noetherian and affine over \(\Lambda\). Perhaps the most important result is a powerful criterion for integrality, reducing the work to checking this for the center of \(S\): If \(S\) is prime, has its center integral over \(R\), and if \(R\) is noetherian modulo its nilradical \(N(R)\), then the trace ring \(T(S)\) is integral over \(R\). This allows several interesting applications to actions of finite groups and of Lie algebras over fields of prime characteristic, presented in the last section. Further topics include a discussion of integrality and embeddings of quotient rings, and an investigation of the relationship between integrality and module- finiteness. Thus, if \(R \subseteq S\) are PI-algebras over a field \(k\) such that \(S\) is affine over \(k\), both \(_ RS\) and \(S_ R\) are finitely generated, and \(\text{PIdeg}(R/P)\geq{1\over 2}\text{PIdeg}(S/N(S))\) for all primes \(P\) of \(R\), then \(S\) is integral over \(R\). This very interesting, substantial paper demonstrates that the notion of Schelter integrality for PI-rings is a much more powerful concept than had been thought before.
scientific article; zbMATH DE number 130648
scientific article