Hostname: page-component-7c8c6479df-995ml Total loading time: 0 Render date: 2024-03-29T11:12:57.930Z Has data issue: false hasContentIssue false

On the minimal prime ideals of a tensor product of two fields

Published online by Cambridge University Press:  24 October 2008

P. Vámos
Affiliation:
University of Sheffield

Extract

Let F be a field, L a commutative F-algebra and K an extension field of F. An important area of commutative algebra is the study of the passage from L to the k-algebra KFL, i.e. the investigation of the behaviour of the ideals of L under ‘extension of scalars’. In most problems of this kind one finds that the problem is reduced to the case when the algebra L is itself an extension field of F. It is for this reason that tensor products of fields play an important role (see, for example, (2), chap, viii, (3), (5), (9) and (12), vol. I).

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Bourbaki, N.Commutative algebra (Reading, Mass., Addison-Wesley, 1972).Google Scholar
(2)Bourbaki, N.Algèbre (Paris, Hermann, 1958/1959).Google Scholar
(3)Cartier, P.Extensions régulières. Séminaire Cartan-Chevalley 8 (1955/1956), 14011410.Google Scholar
(4)Cohn, P. M.Algebra, vol. 2 (London, John Wiley & Sons, 1977).Google Scholar
(5)Grothendieck, A.Élements de géométrie algebrique, vol. iv (Bures-sur-Yvette, Inst. des Hautes Études Sci., Publ. Math. No. 24, 1965).Google Scholar
(6)Jacobson, N.Structure of rings (Providence, Amer. Math. Soc., 1968).Google Scholar
(7)Kaplansky, I.Commutative rings (Chicago, University of Chicago Press, 1974).Google Scholar
(8)Lambek, J.Lectures on rings and modules (Toronto, Blaisdell, 1966).Google Scholar
(9)Lang, S.Introduction to algebraic geometry (Reading, Mass., Addison-Wesley, 1972).Google Scholar
(10)Pierce, R. S.Modules over commutative regular rings. Memoirs Amer. Math. Soc. 70 (1967).Google Scholar
(11)Watanabe, K., Ishikawa, S., Tachibana, S. and Otsuka, K. On tensor products of Gorenstein Rings. J. Math. Kyoto Univ. 9 (1969), 413423.Google Scholar
(12)Zariski, O. and Samuel, S.Commutative algebra, vols. 1, 2 (Princeton, D. van Nostrand, 1958/1960).Google Scholar