TY - JOUR AU - A. Azarang AU - O. A. S. Karamzadeh AU - A. Namazi PY - 2013/07/25 Y2 - 2024/03/29 TI - Hereditary Properties between a Ring and its Maximal Subrings JF - Ukrains’kyi Matematychnyi Zhurnal JA - Ukr. Mat. Zhurn. VL - 65 IS - 7 SE - Research articles DO - UR - https://umj.imath.kiev.ua/index.php/umj/article/view/2475 AB - 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. ER -