  A commutative associative algebra A with an identity over the field of real
numbers which has a basis, where all elements are invertible, is considered in
the work. Moreover, among matrixes consisting of the structure constants of A,
there is at least one that is non-degenerate. The notion of linearly convex
domains in the multi-dimensional complex space and some of their properties are
generalized to the space that is the Cartesian product of n algebras A. Namely,
the separate necessary and sufficient conditions of the local A-linear
convexity of domains with smooth boundary in the space are obtained in terms of
nonnegativity and positivity of formal quadratic differential form in A,
respectively.
