On some counterparts of Rickart $*$-algebras

Abstract

In the present paper, we introduce and study counterparts of Rickart involutive algebras, i.e., almost inner Rickart algebras. We prove that a nilpotent associative algebra, which has no nilpotent elements with nonzero square roots, is an almost inner Rickart algebra. A nilpotent associative algebra, which has no nilpotent elements with a square root $b$ such that $b^3\neq 0$, is not an almost inner Rickart algebra if there exists a nonzero element $a$ such that $a^2\neq 0$. As a main result of the paper, we describe a finite-dimensional almost inner Rickart algebra $\mathcal{A}$ over a field $\mathbb{F}$, isomorphic to $\mathbb{F}^n\dot{+} \mathcal{N}$, $n=1,2$, with a nilradical $\mathcal{N}$. Also, we classify finite-dimensional almost inner Rickart algebras over the real or complex numbers with a nonzero nilradical $\mathcal{N}$.