The notion of deviation of an ordered set has been introduced by Gabriel as a tool to classify rings. It measures how far a given ordered set P deviates from ordered sets satisfying the descending chain condition. We consider here a more general notion and, according to Robson, we define the Krull dimension of P as the deviation of the collection F(P) of its final segments ordered by inclusion....

In this note we show that the quotient field of a domain which is Henselian with respect to a non-trivial ideal is a large field, and give some applications of this fact, using a specialization theorem for ramified covers of the line over (generalized) Krull fields.

The proof of the following theorem is presented: If D is, respectively, a Krull domain, a Dedekind domain, or a PrGfer domain, then D is correspondingly a UFD, a PID, or a Bezout domain if and only if every irreducible quadratic polynomial in D[X] is a prime element.

‎In this article we introduce $mu$-filtered fuzzy module with a family of fuzzy submodules.  It shows the relation between $mu$-filtered fuzzy modules and crisp filtered modules by level sets. We investigate fuzzy topology on the $mu$-filtered fuzzy module and apply that to introduce fuzzy completion. Finally we extend Krull's intersection theorem of fuzzy ideals by using concept $mu$-adic comp...

We prove that the Krull dimension of the ring of holomorphic functions of a connected complex manifold is at least the cardinality of continuum if and only if it is > 0. Let R be a commutative ring. Recall that the Krull dimension dim(R) of R is the supremum of cardinalities lengths of chains of distinct proper prime ideals in R. Our main result is: Theorem 1. Let M be a connected complex manif...