Hereditary Properties between a Ring and its Maximal Subrings

  • A. Azarang
  • O. A. S. Karamzadeh
  • A. Namazi

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
Azarang, A., O. A. S. Karamzadeh, and A. Namazi. “Hereditary Properties Between a Ring and Its Maximal Subrings”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, no. 7, July 2013, pp. 883–893, https://umj.imath.kiev.ua/index.php/umj/article/view/2475.
Section
Research articles