Vol. 58, No. 2, 1975

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Finitely generated modules over Bezout rings

Roger Allen Wiegand and Sylvia Wiegand

Vol. 58 (1975), No. 2, 655–664

Let R be a Bezout ring (a commutative ring in which all finitely generated ideals are principal), and let M be a finitely generated R-module. We will study questions of the following sort: (A) If every localization of M can be generated by n elements, can M itself be generated by n elements? (B) If M RmRn for some m,n, is M necessarily free? (C) If every localization of M has an element with zero annihilator, does M itself have such an element? We will answer these and related questions for various familiar classes of Bezout rings. For example, the answer to (B) is “no” for general Bezout rings but “yes” for Hermite rings (defined below). Also, a Hermlte ring is an elementary divisor ring if and only if (A) has an affirmative answer for every module M.

Mathematical Subject Classification 2000
Primary: 13C05
Received: 23 January 1974
Revised: 16 April 1974
Published: 1 June 1975
Roger Allen Wiegand
Sylvia Wiegand
Department of Mathematics
University of Nebraska-Lincoln
Avery Hall 329
Lincoln NE 68588-0130
United States