We explore several equivalent conditions for finiteness of the set of minimal prime divisors of an ideal and conclude the results of Anderson, Gilmer and Heinzer as especial cases. It is proved that in the ring of polynomials K[X], in which K is a Noetherian ring and X a (possibly infinite) set of indeterminates over K, these conditions are necessary and sufficient. In particular, it is proved that in K[X], an ideal has a finite number of minimal prime divisors if and only if all its minimal prime divisors have finite height. The same results are proved for the derived normal ring of a Noetherian integral domain and the quotient ring K[X]/I, in which I is generated by a K[X]-regular sequence of finite length. We also give a counterexample to show that the conditions of Anderson, Gilmer and Heinzer are not sufficient.

Abstract. Let R be a commutative ring and let M be an R-module. The aim of this paper is to establish an efficient decomposition of a proper submodule N of M as an intersection of primal submodules. We prove the existence of a canonical primal decomposition, where the intersection is taken over the isolated components N (p) of N that are primal submodules having distinct and incomparable adjoint prime ideals p. Using this decomposition, we prove that for p ∈ Supp(M/N ), the submodule N is an intersection of p-primal submodules if and only if the elements of R \ p are prime to N . Also, it is shown that M is an arithmetical R-module if and only if every primal submodule of M is irreducible. Finally, we determine conditions for the canonical primal decomposition to be irredundant or residually maximal, and for the unique decomposition of N as an irredundant intersection of isolated components.1. Introduction. Throughout this paper, all rings considered will be commutative and will have non-zero identity elements and all modules will be unitary. Such a ring will be denoted by R, and the terminology is, in general, the same as that in [1] and [6]. Associated primes and primary decompositions are the most basic notions in the study of modules over commutative Noetherian rings. However, for modules over non-Noetherian rings the classical associated primes and primary decompositions are also interesting (see [7] and [8]). Let M be an R-module and N a proper submodule of M . In [13] (resp. [16]) Krull (resp. Noether) has introduced the most useful concept of associated primes (resp. primary decomposition) of N .A prime ideal p of R is said to be a Krull associated prime of the submodule N if for every element x ∈ p, there exists m ∈ M such that x ∈ N : R m ⊆ p. A prime ideal p of R is called a weakly (resp. ZariskiSamuel ) associated prime to N if there exists an element m ∈ M such that p is minimal over the annihilator N : R m (resp. p = Rad(N : R m)). We will denote the set of Krull (resp. weakly) associated primes to N by Ass R M/N

No abstract

UDC 512.5 Let be a homogeneous Noetherian ring with semilocal base ring Let be the irrelevant ideal of For two finitely generated graded -modules and several results on the vanishing, Artiniannes and tameness property of the graded -modules will be investigated.

No abstract