# Hereditary Properties between a Ring and its Maximal Subrings

### Abstract

We study the existence of maximal subrings and hereditary properties between a ring and its maximal subrings. Some new techniques for establishing the existence of maximal subrings are presented. It is shown that if*R*is an integral domain and

*S*is a maximal subring of

*R*, then the relation dim(

*R*) = 1 implies that dim(

*S*) = 1 and vice versa if and only if (

*S*:

*R*) = 0. Thus, it is shown that if

*S*is a maximal subring of a Dedekind domain

*R*integrally closed in

*R*; then

*S*is a Dedekind domain if and only if

*S*is Noetherian and (

*S*:

*R*) = 0. We also give some properties of maximal subrings of one-dimensional valuation domains and zero-dimensional rings. Some other hereditary properties, such as semiprimarity, semisimplicity, and regularity are also studied.

Published

25.07.2013

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 65, no. 7, July 2013, pp. 883–893, http://umj.imath.kiev.ua/index.php/umj/article/view/2475.

Issue

Section

Research articles