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 Γ is the centroid of a prime algebra A and Λ is the field of fractions of Γ
then (under mild hypotheses) A ⊗ΓΛ is shown to have Λ 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 Φ and F is an
extension field of Φ, then A ⊗ΦF 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.
