Noncommutative Prüfer rings satisfying a polynomial identity
A prime Goldie ring \(R\) is a right Prüfer ring if for every finitely generated right \(R\)-ideal \(I\), \(I^{-1}I = R\) and \(II^{-1} = O_ l(I)\). \(R\) is a left Prüfer ring if for every finitely generated left \(R\)-ideal \(J\) of \(R\), \(JJ^{-1} = R\) and \(J^{-1}J = O_ r(R)\). \textit{J. H. Alajbegovic} and \textit{N. I. Dubrovin} [J. Algebra 135, 165-176 (1990; Zbl 0718.16023)] have shown that a ring is right Prüfer if and only if it is left Prüfer. The paper investigates the relation of Prüfer rings to other types of rings as e.g. valuation rings (in the sense of Dubrovin) and Bezout rings (rings in which every finitely generated right ideal is principal). The main theorems are: (Theorem 1.9) Let \(R\) be Prüfer and \(B_ 1,\dots,B_ n\) be valuation (Dubrovin) overrings of \(R\) then the \(B_ i\) satisfy the intersection property, i.e. \(Q \mapsto Q \cap R\) gives a one-to-one correspondence between the prime ideals of the various \(B_ i\) and the prime ideals of \(R\). (Theorem 2.5) Let \(R\) be a prime PI-ring with center \(A\). Suppose \(A\) is Prüfer. Then \(R\) is a Prüfer ring in which each maximal ideal is localizable if and only if \(R_ m\) is Bezout for all maximal ideals \(m\) of \(A\).
scientific article; zbMATH DE number 554245