A Note on the n-Generator Property for Commutative Monoids |
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Authors: | Greg Oman |
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Affiliation: | (1) Department of Mathematics, The Ohio State University, 231 W. 18th Ave., Columbus, OH 43210, USA |
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Abstract: | Let M be a cancellative, commutative monoid with integral closure . Borrowing from ring theory, we say that M has the n-generator property iff every finitely generated ideal of M can be generated by n elements, and we say M has rank n iff every ideal of M can be generated by n elements. We investigate the integral closure of such monoids. We show, in particular, that if M has the n-generator property, then is a valuation monoid, and if M has rank n, then is a principal ideal monoid. |
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