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The centre of a hereditary local ring

Published online by Cambridge University Press:  18 May 2009

D. G. Northcott
D. G. Northcott
Affiliation:
The University, Sheffield
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The purpose of this note is to establish the following

Theorem. 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.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 5 , Issue 3 , January 1962 , pp. 101 - 102
Copyright
Copyright © Glasgow Mathematical Journal Trust 1962

References

1.Cartan, H. and Eilenberg, S., Homological algebra (Princeton, 1956).Google Scholar
2.Kaplansky, I., Projective modules, Ann. of Math. 68 (1958), 372–377.CrossRefGoogle Scholar