  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
$\mathcal{N}$.
