Year: 2025
Type: article
Abstract: In this paper, we study maximal subrings up to isomorphism of fields. It is shown that each field with zero characteristic has infinitely many maximal subrings up to isomorphism. If [Formula: see text] is an algebraically closed field, [Formula: see text] is an indeterminate over [Formula: see text] and [Formula: see text] and [Formula: see text] are integrally closed maximal subrings of [Formula: see text] which contain [Formula: see text], then we prove that [Formula: see text]. In particular, if [Formula: see text] is an absolutely algebraic field, then [Formula: see text] has only finitely many integrally closed maximal subrings up to isomorphism if and only if [Formula: see text] is algebraically closed. Also, we show that if [Formula: see text] is an absolutely algebraic field then [Formula: see text] has only finitely many maximal subrings up to isomorphism if and only if [Formula: see text] has only finitely many maximal subrings. We prove that if a commutative ring [Formula: see text] with zero characteristic has only finitely many maximal subrings up to isomorphism, then [Formula: see text] is finite, where [Formula: see text] is the set of natural prime numbers. In particular, if [Formula: see text] is a commutative ring with zero characteristic and [Formula: see text], then [Formula: see text] has infinitely many maximal subrings up to isomorphism. If [Formula: see text] is a field, then we investigate the maximal subrings up to isomorphism of [Formula: see text], [Formula: see text] and [Formula: see text]. If [Formula: see text] is a non-field maximal subring of a field [Formula: see text] and [Formula: see text] is the set of all maximal subrings of [Formula: see text] which are isomorphic to [Formula: see text], then we prove that [Formula: see text], in particular [Formula: see text]. Moreover if, [Formula: see text] is infinite and [Formula: see text], then [Formula: see text] has at least [Formula: see text]-many integrally closed maximal subrings up to isomorphism.
Authors: Alborz Azarang, Nasrin Parsa
Institution: Shahid Chamran University of Ahvaz
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4408676893