Non-commutative regular local rings of dimension 2
Let P be a prime ideal of the noetherian ring R. The set of nonzero divisors modulo P, \({\mathcal C}(P)\), is not usually an Ore set, but contains a maximal Ore set S(P). There is now a general theory concerning the localization theory at a prime ideal in a noetherian ring and this is presented in the recent book by \textit{A. V. Jategaonkar} [Localization in Noetherian rings (1986; Zbl 0589.16014)]. In this paper the author considers in detail a very special case. Specifically, suppose that \(P=xR+yR=Rx+Ry\) where \(Ry=yR\), y is a nonzero divisor and x is a nonzero divisor modulo yR. For technical reasons one demands that R contain an uncountable central subfield. The maximal Ore set S(P) is identified as the elements that are nonzero divisors modulo each prime of the form \(y^{-n}Py\) n. The resulting Ore localization \(T=RS^{-1}\) is then studied. The ring T is prime, although the ring T \(*=T/yT\) need only be semiprime. The ring T has Krull and global dimensions equal to two and the maximal ideals of T are induced by \(y^{-n}Py\) n. T * is a direct sum of bounded hereditary noetherian prime rings in which every two-sided ideal is principal. The number n of prime rings in this direct sum has a great influence on the structure of T. For example, if \(n=1\) then T is an integral domain over which every finitely generated projective module is free, while if n is prime and T is not an integral domain then T is an \(n\times n\) tiled matrix ring over a domain D with similar properties to T, but \(n(D)=1\).
direct sum of bounded hereditary noetherian prime rings
scientific article; zbMATH DE number 4057715
scientific article