Remarks on Blowing-Up Divisorial Ideals.
A ring extension is said to be FO if it has only finitely many intermediate rings. is said to be FC if each chain of distinct intermediate rings in this extension is finite. We establish several necessary and sufficient conditions for the ring extension to be FO or FC together with several other finiteness conditions on the set of intermediate rings. As a corollary we show that each integrally closed ring extension with finite length chains of intermediate rings is necessarily a normal pair...
We introduce and study a new class of ring extensions based on a new formula involving the heights of their primes. We compare them with the classical altitude inequality and altitude formula, and we give another characterization of locally Jaffard domains, and domains satisfying absolutely the altitude inequality (resp., the altitude formula). Then we study the extensions R ⊆ S where R satisfies the corresponding condition with respect to S (Definition 3.1). This leads to a new characterization...
Let be a commutative Noetherian ring, an ideal of , an -module and a non-negative integer. In this paper we show that the class of minimax modules includes the class of modules. The main result is that if the -module is finite (finitely generated), is -cofinite for all and is minimax then is -cofinite. As a consequence we show that if and are finite -modules and is minimax for all then the set of associated prime ideals of the generalized local cohomology module...
Let be a commutative Noetherian ring with identity and an ideal of . It is shown that, if is a non-zero minimax -module such that for all , then the -module is -cominimax for all . In fact, is -cofinite for all . Also, we prove that for a weakly Laskerian -module , if is local and is a non-negative integer such that for all , then and are weakly Laskerian for all and all . As a consequence, the set of associated primes of is finite for all , whenever and...