Almost splitting sets in integral domains, II |
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Authors: | David F. Anderson |
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Affiliation: | a Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, USA b Department of Mathematics, University of Incheon, Incheon 402-749, Republic of Korea |
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Abstract: | Let D be an integral domain. A saturated multiplicative subset S of D is an almost splitting set if, for each 0≠d∈D, there exists a positive integer n=n(d) such that dn=st for some s∈S and t∈D which is v-coprime to each element of S. We show that every upper to zero in D[X] contains a primary element if and only if D?{0} is an almost splitting set in D[X], if and only if D is a UMT-domain and Cl(D[X]) is torsion. We also prove that D[X] is an almost GCD-domain if and only if D is an almost GCD-domain and Cl(D[X]) is torsion. Using this result, we construct an integral domain D such that Cl(D) is torsion, but Cl(D[X]) is not torsion. |
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Keywords: | 13A15 13B25 13F05 13F20 13G05 |
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