Connected graded Gorenstein algebras with enough normal elements
Let \(A=k\oplus A_1\oplus A_2\oplus\cdots\) be a connected graded algebra. The algebra \(A\) is said to have enough normal elements if for every graded prime ideal \(P\) of \(A\) the factor ring \(A/P\) contains a nonzero homogeneous normal element of degree \(\geq 1\). The author proves the following theorem which generalises earlier work of \textit{J. T. Stafford} and the author [J. Algebra 168, No. 3, 988-1026 (1994; Zbl 0812.16046)]. Theorem: Let \(A\) be a connected graded Noetherian algebra of injective dimension \(d\) with enough normal elements. Then (i) \(A\) is Auslander-Gorenstein and Cohen-Macaulay, (ii) \(A\) has a quasi-Frobenius quotient ring, (iii) every minimal prime ideal \(P\) is graded and \(\text{GKdim}(A/P)=d\), (iv) if, moreover, \(A\) has finite global dimension, then \(A\) is a domain and a maximal order in its quotient division ring. The earlier work of Stafford and Zhang applies to PI rings, which certainly have enough normal elements. Ample justification for this generalization is provided by the fact that many quantum algebras, while not being PI rings nevertheless have enough normal elements [see \textit{K. R. Goodearl} and the reviewer, J. Pure Appl. Algebra 111, No.~1-3, 123-142 (1996; Zbl 0864.16018)].
scientific article; zbMATH DE number 1013991
scientific article