Symmetry and localization in Noetherian prime PI rings
Let \(R\) be a prime Noetherian P.I. ring, and let \(T(R)\) be its trace ring (so \(T(R)\) is the subring of the quotient ring of \(R\) got by adjoining to \(R\) the coefficients of the Cayley-Hamilton polynomials of the elements of \(R\), and is a finite central extension of \(R\)). Let \(P\) and \(Q\) be prime ideals of \(R\), and say that \(P\) and \(Q\) are tr-linked if there exist prime ideals \(P_ 1\) and \(Q_ 1\) of \(T(R)\) with \(P_ 1\cap R=P\), \(Q_ 1\cap R=Q\), and \(P_ 1\) and \(Q_ 1\) having the same intersection with the centre of \(T(R)\). The main theorem of this paper gives an equivalence between the 3 statements: (i) \(P\) is right (Ore) localizable; (ii) \(P\) is left localizable; and (iii) if \(Q\) is tr-linked to \(P\), then \(Q=P\). This result is, in part, a special case of a result relating the graphs of links of \(R\), and of \(T(R)\). This aspect, and its connections to the representation theory of \(R\), have been taken further in [\textit{T. H. Lenagan} and \textit{E. Letzter}, The fundamental prime ideals of a Noetherian prime P.I. ring, Proc. Edinb. Math. Soc., II. Ser. 33, No. 1, 113--121 (1990; Zbl 0697.16016)].
scientific article; zbMATH DE number 4075310
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