ИСТИНА

By the length of a finite system of generators for a finite-dimensional algebra over an arbitrary field we mean the least positive integer k such that the products of length not exceeding k span this algebra (as a vector space). The maximum length for the systems of generators of an algebra is referred to as the length of the algebra. The length evaluation can be a difficult problem, for example, the length of the full matrix algebra is unknown (Paz’s Problem, 1984). We examine some basic properties of the length function, including the length of a direct sum of algebras and the length of an algebra after the field extension, and relationship of this characteristic with other numerical invariants of the given algebra. The main topic of this talk is the length of commutative algebras. We give an exact upper bound for the length of a commutative algebra in general and provide more specific bounds and exact length computation for commutative subalgebras in the matrix algebra and for group algebras of finite Abelian groups.