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Some Remarks on Non-Commutative Extensions of Local Rings

Published online by Cambridge University Press:  22 January 2016

Edward H. Batho
Edward H. Batho*
Affiliation:
University of Rochester
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In [1] we introduced the concept of a non-commutative local ring and studied the structure of such rings. Unfortunately, we were not able to show that the completion of a local ring was a semi-local ring. In this paper we propose to study a class of rings for which the above result is valid. This class of rings is the integral extensions -[4, 5]-of commutative local rings. This class of rings includes the important class of matrix rings over commutative local rings. In part 1 below we study some elementary properties of integral extensions and here we assume merely that the underlying ring is semi-local. In part 2 we discuss some questions of ideal theory for arbitrary local rings as well as for integral extensions. In a later paper we propose to utilize our results to study the deeper properties of these rings including a dimension theory for such rings. We are particularly indebted to the work of Nagata [6, 7, 8, 9] in the preparation of this paper.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 14 , February 1959 , pp. 45 - 51
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1959

References

[1] Batho, E. H., Non-commutative semi-local and local rings, Duke Math. Journal, vol. 85 (1957), pp. 163–172.Google Scholar
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[9] Nagata, M., On the theory of semi-local rings. Proceedings of the Japan Academy, vol. 85 (1950), pp. 131–140.Google Scholar