# 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.

**English version** (Springer):
Ukrainian Mathematical Journal **65** (2013), no. 7, pp 981-994.

**Citation Example:** *Azarang A., Karamzadeh O. A. S., Namazi A.* Hereditary Properties between a Ring and its Maximal Subrings // Ukr. Mat. Zh. - 2013. - **65**, № 7. - pp. 883–893.

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