Vol. 94, No. 1, 1981

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Conditions for being an FGC domain

Willy Brandal

Vol. 94 (1981), No. 1, 1–12

A domain R is said to be FGC if every finitely generated R-module decomposes into a direct sum of cyclic submodules. The main result is: if R is a domain with quotient field Q, then R is FGC if and only if all of the following three conditions are satisfied: (1) R is Bezout, (2) Q∕R is an injective R-module, and (3) there does not exist a continuous embedding of βN into spec R relative to the patch topology of spec R. This result is also true if (3) is replaced: (3) every nonzero element of R is an element of only finitely many maximal ideals of R. Using entire functions, there exists an example of a domain satisfying (1) and (2), but not satisfying (3). Also presented are some partial results towards generalizing the main result to commutative rings.

Mathematical Subject Classification 2000
Primary: 13C05
Secondary: 13F05
Received: 8 April 1980
Published: 1 May 1981
Willy Brandal