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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Flat modules over commutative noetherian rings

Author: Wolmer V. Vasconcelos
Journal: Trans. Amer. Math. Soc. 152 (1970), 137-143
MSC: Primary 13.40; Secondary 16.00
MathSciNet review: 0265341
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Abstract: In this work we study flat modules over commutative noetherian rings under two kinds of restriction: that the modules are either submodules of free modules or that they have finite rank. The first ones have nicely behaved annihilators: they are generated by idempotents. Among the various questions related to flat modules of finite rank, emphasis is placed on discussing conditions implying its finite generation, as for instance, (i) over a local ring, a flat module of constant rank is free, and (ii) a flat submodule of finite rank of a free module is finitely generated. The rank one flat modules already present special problems regarding its endomorphism ring; in a few cases it is proved that they are flat over the base ring. Finally, a special class of flat modules--unmixed--is discussed, which have, so to speak, its source of divisibility somewhat concentrated in the center of its endomorphism ring and thus resemble projective modules over flat epimorphic images of the base ring.

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Keywords: Flat modules, integrally closed noetherian domains, flat overrings, unmixed modules
Article copyright: © Copyright 1970 American Mathematical Society

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