MODULES HAVING VERY FEW ZERO-DIVISORS

Abstract

Let R be a commutative ring, I a finitely generated ideal of R, and M a zero-divisor R-module. It is shown that the M-grade of I defined by the Koszul complex is consistent with the definition of M-grade of I defined by the length of maximal M-sequences in I. Also, it is shown that, if R is Noetherian, then for any submodule N of M, the following conditions are equivalent: (1) Ass(R)(H(I)(0)(M/N)/L) is finite for all finitely generated submodules L of H(I)(0)(M/N); (2) If (H(I)(0)(M/N)/L), ... ,H(I)(i-1)(M/N) are finitely generated, then Ass(R)(H(I)(i)(M/N)/L) is finite for all finitely generated submodules L of H(I)(i)(M/N).