Prime nonassociative algebras

Abstract

An arbitrary algebra (not necessarily associative or unital) is said to be prime if the product of any two nonzero ideals is nonzero. The hypothesis that an algebra is prime has now been used in the study of several different varieties of nonassociative algebras, and the need for an understanding of the basic properties of prime nonassociative algebras has become apparent. If is the centroid of a prime algebra A and is the field of fractions of then (under mild hypotheses) A is shown to have as its centroid. The extended centroid C of a prime algebra A can be defined, the central closure Q of A can be constructed, and Q is shown to be closed in the sense that it is its own central closure. Tensor products are studied and among other results the following are obtained: (1) if A is a closed prime algebra over and F is an extension field of , then A(g)F is a closed prime algebra over F, (2) the tensor product of closed prime algebras is closed. Finally, the results on prime algebras are specialized to obtain results on the tensor products of simple algebras.