GK-dimensions of corners and ideals
Let R be a prime algebra over a field k, and let e be an idempotent element of R. If R is either PI or noetherian, then it is known (see, for example, \textit{S. Montgomery} [Contemp. Math. 88, 103-124 (1989; Zbl 0675.16018)]) that \(GK(eRe)=GK(R)\), where GK(-) denotes the Gelfand- Kirillov dimension, and the problem remains open for the prime Goldie case. The authors give an example to show that in a general prime ring GK(eRe) may be any integer \(<GK(R)\). The key result for the analysis is that for a corner (also called subideal) of R, that is, a subalgebra T with TRT\(\subseteq T\), one always has \(GK(T)=GK(RT^ 2R)\). The above- stated problem is closely connected with the problem whether \(GK(R)=GK(A)\), as an algebra, for any nonzero ideal A of a prime k- algebra R. In fact, for prime Goldie rings the two questions are equivalent, meaning that a prime Goldie ring R exists with a nonzero ideal A such that \(GK(A)<GK(R)\), if and only if a prime Goldie ring S exists with an idempotent e such that \(GK(eSe)<GK(S)\). On the positive side, it is shown that if R is prime Goldie with GK(R)\(\leq 2\), then indeed \(GK(A)=GK(R)\) for any ideal \(A\neq 0\).
scientific article; zbMATH DE number 4142254
scientific article