Representational repleteness of differential operator rings
Let \(P\) be a prime ideal of Noetherian ring \(R\), and let \(E\) be an indecomposable injective right \(R\)-module with assassinator \(P\). Then \(P\) is said to be (right) representationally replete (rep. rep.) if every prime ideal which could be involved in \(E\) is involved, in the following sense: Let \(A_n(P)\) be the set of associated primes of the \(n\)th layer of \(E\) and let \(X_n(P)\) be the set of primes which are \((n-1)\) second layer links away from \(P\) in the graph of links. Then \(P\) is right rep. rep. if \(\bigcup^\infty_{n=1}A_n(P)=\bigcup^\infty_{n=1}X_n(P)\); it is right strongly rep. rep. if \(A_n(p)=X_n(p)\) for all \(n\). (These concepts are independent of the choice of \(E\), as shown in [\textit{K. A. Brown} and \textit{R. B. Warfield} jun., J. Algebra 116, 294-315 (1988; Zbl 0652.16006)], where the terms used above are also defined.) In the paper under review, which is a sequel to [\textit{G. M. Low}, Injective modules and representational repleteness, J. Algebra (to appear)], the case where \(R\) is a ring of differential operators is examined in detail. A sample result is the following. Let \(R=S[\theta_1,\dots,\theta_n;\delta_1,\dots,\delta_n]\) be a differential operator ring, where \(S\) is a commutative Noetherian \(\mathbb{Q}\)-algebra and \(\delta_1,\dots,\delta_n\) are commuting \(\mathbb{Q}\)-derivations of \(S\). Let \(P\) be a prime ideal of \(R\), and set \(M=P\cap S\). If \(S_M\) is a regular local ring then \(P\) is rep. rep.
scientific article; zbMATH DE number 954741
scientific article