Hereditary Properties between a Ring and its Maximal Subrings
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.