Journal article
A Note on Almost GCD Monoids
Semigroup Forum, Vol.69(1), pp.141-154
06/2004
DOI: 10.1007/s00233-004-0119-z
Abstract
A commutative cancellative monoid H (with 0 adjoined) is called an almost GCD (AGCD) monoid if for x,y in H, there exists a natural number n = n(x,y) so that xn and yn have an LCM, that is, xnH \cap ynH is principal. We relate AGCD monoids to the recently introduced inside factorial monoids (there is a subset Q of H so that the submonoid F of H generated by Q and the units of H is factorial and some power of each element of H is in F). For example, we show that an inside factorial monoid H is an AGCD monoid if and only if the elements of Q are primary in H, or equivalently, H is weakly Krull, distinct elements of Q are v-coprime in H, or the radical of each element of Q is a maximal t-ideal of H. Conditions are given for an AGCD monoid to be inside factorial and the results are put in the context of integral domains.
Details
- Title: Subtitle
- A Note on Almost GCD Monoids
- Creators
- D.D Anderson - Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419 USAMuhammad Zafrullah - Department of Mathematics, Idaho State University, Pocatello, ID 83209-8085 USA
- Resource Type
- Journal article
- Publication Details
- Semigroup Forum, Vol.69(1), pp.141-154
- DOI
- 10.1007/s00233-004-0119-z
- ISSN
- 0037-1912
- eISSN
- 1432-2137
- Publisher
- Springer-Verlag; New York
- Language
- English
- Date published
- 06/2004
- Academic Unit
- Mathematics
- Record Identifier
- 9983985947902771
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