Ranks of tensors and change of base field

AbstractIf M is a subspace of a tensor product of vector spaces, A ⊕ B, we define r(M) = inf rank x(x ϵ M — {0}). (Several characterizations of the rank of a tensor are recalled in Section 1.) The main point of this note is to introduce this function r, and establish its basic properties under change of base field. We were originally led to the subject by a problem in the construction of some rather esoteric noncommutative rings ([1], Section 7), but in Section 3 we shall indicate a relation with the study of finite-dimensional algebras over a field. Perhaps this function will be found useful in still other areas