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A Remark on Flat and Projective Modules

Published online by Cambridge University Press:  20 November 2018

Chr. U. Jensen
Chr. U. Jensen*
Affiliation:
University of Copenhagen, Denmark
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It is the purpose of this note to give some characterizations of flat and projective modules, partly in ideal theoretical terms, partly in terms of the exterior product of a module (“puissance extérieure“); cf. (1).

We shall consider left modules over a ring R with identity element and without proper zero divisors. The left module M is called flat if X ⊗R M is an exact functor on the category of right R-modules X. If M is flat over a commutative domain R, M is necessarily torsion-free. Therefore when looking for flatness of a module M over a commutative domain, one may assume from the start that M is torsion-free.

In the following theorem, we shall not restrict ourselves to commutative rings R, but the modules concerned have to be torsion-free, which, of course, should mean that rm = 0 implies r = 0 or m = 0.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 943 - 949
Copyright
Copyright © Canadian Mathematical Society 1966

References

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