When is the tensor product of algebras local?

Sweedler, Moss Eisenberg

doi:10.1090/S0002-9939-1975-0360568-6

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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When is the tensor product of algebras local?
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by Moss Eisenberg Sweedler PDF

Proc. Amer. Math. Soc. 48 (1975), 8-10 Request permission

Abstract:

Suppose the tensor product of two commutative algebras over a field is local. It is easily shown that each of the commutative algebras is local and that the tensor product of the residue fields is local. Moreover, one of the algebras must be algebraic over the ground field, i.e. contain no transcendentals. These three conditions characterize when the tensor product of commutative algebras is local.

References

S. A. Amitsur, The radical of field extensions, Bull. Res. Council Israel Sect. F 7F (1957/58), 1–10. MR 103912
N. Jacobson, The radical and semi-simplicity for arbitrary rings, Amer. J. Math. 67 (1945), 300–320. MR 12271, DOI 10.2307/2371731
Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
Moss Eisenberg Sweedler, A units theorem applied to Hopf algebras and Amitsur cohomology, Amer. J. Math. 92 (1970), 259–271. MR 268255, DOI 10.2307/2373506

Additional Information

Journal: Proc. Amer. Math. Soc. 48 (1975), 8-10
DOI: https://doi.org/10.1090/S0002-9939-1975-0360568-6
MathSciNet review: 0360568