Conjugacy classes of left ideals of a finite dimensional algebra

Abstract

Let A be a finite dimensional unital algebra over a field K and let C(A) denote the set of conjugacy classes of left ideals in A. It is shown that C(A) is finite if and only if the number of conjugacy classes of nilpotent left ideals in A is finite. The set C(A) can be considered as a semigroup under the natural operation induced from the multiplication in A. If K is algebraically closed, the square of the radical of A is zero and C(A) is finite, then for every K-algebra B such that C(B) ≡ C(A) it is shown that B ≡ A.