Assume you've got a set $S$ and an associative multiplication $S\times S\to S$. Let's further assume that for some given $a\in S$ there exists an element $b\in S$ such that $a^2b = aba = ba^2 = a$. Now define two elements $n=ab$ and $i=ab^2$ (note that $n$ resembles somewhat a neutral element for $a$, and $i$ resembles somewhat an inverse element, however they may be different for different elements, and $S$ might not have a neutral element at all). My question is: Do those elements $n$ and $i$ have established names? Also, does there exist a name for those elements $a$ for which such a $b$ (and thus an $n$ and an $i$) exists?
BTW, note that while there can be different elements $b$ with this property for a given $a$, $n$ and $i$ do not depend on the choice of $b$ (but in general will depend on the given $a$).
(I'm sorry for the bad title, but I can't think of a better one; feel free to improve it.)