On noncommutative localization
Let R be a ring which for convenience will be assumed to be right and left Noetherian although this is not always necessary, and let P, S, N denote respectively a prime ideal, a semi-prime ideal, and the nilpotent radical of R. The two localizations considered are: the local ring Q of R at P as constructed by Goldie; and the inversive localization of R at S which in this review will be denoted by R(S), i.e. R(S) is universal with respect to the property that all square matrices over R which are regular modulo S are invertible over the ring R(S). It is shown that R(S) is Artinian if and only if some power of S is contained in the kernel of the natural map from R to R(S), and this condition is satisfied if \(S=N\). Also Q is isomorphic to R(P)/J' where J' is the intersection of the powers of the Jacobson radical of R(P); and Q is universal with respect to the properties (1) Q/J(Q) is naturally isomorphic to the quotient ring of R/P, and (2) the intersection of the powers of J(Q) is zero.
scientific article; zbMATH DE number 3867521
scientific article