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 $C(B)\cong C(A)$ it is shown that $B\cong A$.