The centre of a hereditary local ring
Glasgow mathematical journal, Tome 5 (1962) no. 3, pp. 101-102
Voir la notice de l'article provenant de la source Cambridge University Press
The purpose of this note is to establish the followingTheorem. The centre of a (left) hereditary local ring is either afield or a one-dimensional regular local ring.Before starting the proof, it is necessary to explain the terminology. A ring R with an identity element is called a left local ring if the elements of R which do not have left inverses form a left ideal I. In these circumstances (see [1, Proposition 2.1, p. 147]), I is necessarily a two-sided ideal and it consists precisely of all the elements of R which do not have right inverses. Furthermore, every element of R which is not in I possesses a two-sided inverse. Thus there is, in fact, no difference between a left local ring and a right local ring and therefore one speaks simply of a local ring. In addition, I contains every proper left ideal and every proper right ideal. We may therefore describe I simply as the maximal ideal of R.
Northcott, D. G. The centre of a hereditary local ring. Glasgow mathematical journal, Tome 5 (1962) no. 3, pp. 101-102. doi: 10.1017/S2040618500034407
@article{10_1017_S2040618500034407,
author = {Northcott, D. G.},
title = {The centre of a hereditary local ring},
journal = {Glasgow mathematical journal},
pages = {101--102},
year = {1962},
volume = {5},
number = {3},
doi = {10.1017/S2040618500034407},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S2040618500034407/}
}
[1] 1.Cartan, H. and Eilenberg, S., Homological algebra (Princeton, 1956). Google Scholar
[2] 2.Kaplansky, I., Projective modules, Ann. of Math. 68 (1958), 372–377. Google Scholar | DOI
Cité par Sources :