Some countability conditions on commutative ring extensions

Gilmer, Robert; Heinzer, William

doi:10.1090/S0002-9947-1981-0597878-0

Some countability conditions on commutative ring extensions
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by Robert Gilmer and William Heinzer PDF

Trans. Amer. Math. Soc. 264 (1981), 217-234 Request permission

Abstract:

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

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