On tensor products and extended centroids

Nicholson, W. K.; Watters, J. F.

doi:10.1090/S0002-9939-1983-0695244-8

On tensor products and extended centroids
HTML articles powered by AMS MathViewer

by W. K. Nicholson and J. F. Watters PDF

Proc. Amer. Math. Soc. 88 (1983), 215-217 Request permission

Abstract:

For prime algebras $R$ and $S$ over a field $F$ it is shown that each nonzero ideal of $R \otimes S$ contains a nonzero element $r \otimes s$, $r \in R$, $s \in S$, if and only if $C(R) \otimes C(S)$ is a field, where $C(R)$ (respectively $C(S)$) is the extended centroid of $R$ (respectively $S$).

References

S. A. Amitsur, On rings of quotients, Symposia Mathematica, Vol. VIII (Convegno sulleAlgebre Associative, INDAM, Rome, 1970) Academic Press, London, 1972, pp. 149–164. MR 0332855
Theodore S. Erickson, Wallace S. Martindale 3rd, and J. Marshall Osborn, Prime nonassociative algebras, Pacific J. Math. 60 (1975), no. 1, 49–63. MR 382379
J. Krempa, On semisimplicity of tensor products, Ring theory (Proc. Antwerp Conf. (NATO Adv. Study Inst.), Univ. Antwerp, Antwerp, 1978) Lecture Notes in Pure and Appl. Math., vol. 51, Dekker, New York, 1979, pp. 105–122. MR 563289
Wallace S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576–584. MR 238897, DOI 10.1016/0021-8693(69)90029-5
J. Matczuk, Central closure of semiprime tensor products, Comm. Algebra 10 (1982), no. 3, 263–278. MR 651603, DOI 10.1080/00927878208822715
Richard Resco, A reduction theorem for the primitivity of tensor products, Math. Z. 170 (1980), no. 1, 65–76. MR 558888, DOI 10.1007/BF01214712