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Radical Regularity in Differential Rings

Published online by Cambridge University Press:  20 November 2018

Howard E. Gorman
Howard E. Gorman*
Affiliation:
Stanford University, Stanford, California
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In [1], we discussed completions of differentially finitely generated modules over a differential ring R. It was necessary that the topology of the module be induced by a differential ideal of R and it was natural that this ideal be contained in J(R), the Jacobson radical of R. The ideal to be chosen, called Jd(R), was the intersection of those ideals which are maximal among the differential ideals of R. The question as to when Jd(R) ⊆ J(R) led to the definition of a class of rings called radically regular rings. These rings do satisfy the inclusion, and we showed in [1, Theorem 2] that R could always be “extended”, via localization, to a radically regular ring in such a way as to preserve all its differential prime ideals.

In the present paper, we discuss the stability of radical regularity under quotient maps, localization, adjunction of a differential indeterminate, and integral extensions.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 2 , April 1971 , pp. 197 - 201
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Gorman, H. E., Differential rings and modules (to appear in Scripta Math.).Google Scholar
2. Kaplansky, I., An introduction to differential algebra, Actualités Sci. Indust., No. 1251 = Publ. Inst. Math. Univ. Nancago, No. 5 (Hermann, Paris, 1957).Google Scholar