95 research outputs found
Computing the closure of a support
When is an -module over a commutative unital ring , the Zariski
closure of its support is of the form where
is a unique radical ideal. We give an explicit form of
and study its behavior under various operations of algebra.
Applications are given, in particular for ring extensions of commutative unital
rings whose supports are closed. We provide some applications to crucial and
critical ideals of ring extensions
Distributive FCP extensions
We are dealing with extensions of commutative rings whose
chains of the poset of their subextensions are finite ({\em i.e.}
has the FCP property) and such that is a distributive
lattice, that we call distributive FCP extensions. Note that the lattice
of a distributive FCP extension is finite. This paper is the
continuation of our earlier papers where we studied catenarian and Boolean
extensions. Actually, for an FCP extension, the following implications hold:
Boolean distributive catenarian. A comprehensive
characterization of distributive FCP extensions actually remains a challenge,
essentially because the same problem for field extensions is not completely
solved. Nevertheless, we are able to exhibit a lot of positive results for some
classes of extensions. A main result is that an FCP extension is
distributive if and only if is distributive, where
is the integral closure of in . A special attention is
paid to distributive field extensions
- β¦