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.
Jensen, Chr. U. A Remark on Flat and Projective Modules. Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 943-949. doi: 10.4153/CJM-1966-093-7
@article{10_4153_CJM_1966_093_7,
author = {Jensen, Chr. U.},
title = {A {Remark} on {Flat} and {Projective} {Modules}},
journal = {Canadian journal of mathematics},
pages = {943--949},
year = {1966},
volume = {18},
number = {1},
doi = {10.4153/CJM-1966-093-7},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1966-093-7/}
}
TY - JOUR
AU - Jensen, Chr. U.
TI - A Remark on Flat and Projective Modules
JO - Canadian journal of mathematics
PY - 1966
SP - 943
EP - 949
VL - 18
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1966-093-7/
DO - 10.4153/CJM-1966-093-7
ID - 10_4153_CJM_1966_093_7
ER -
%0 Journal Article
%A Jensen, Chr. U.
%T A Remark on Flat and Projective Modules
%J Canadian journal of mathematics
%D 1966
%P 943-949
%V 18
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1966-093-7/
%R 10.4153/CJM-1966-093-7
%F 10_4153_CJM_1966_093_7
