Some countability conditions on commutative ring extensions

Abstract

If S S is a finitely generated unitary extension ring of the commutative ring R R , then S S cannot be expressed as the union of a strictly ascending sequence { R n } n = 1 ∞ \{ {R_n}\} _{n = 1}^\infty of intermediate subrings. A primary concern of this paper is that of determining the class of commutative rings T T for which the converse holds—that is, each unitary extension of T T not expressible as ∪ 1 ∞ T i \cup _1^\infty {T_i} is finitely generated over T T .