What do we call the inverse of a right-multiplication when it's the left-multiplication of the inverse element?

Question

I'm working with a non-commutative, non-associative, non-unital algebra with objects that are somewhat like matrices, but have the property that for almost all $X$ (except the "zero-like" elements), there exists a unique element $X'$ such that for all $A$, $X'AX = XAX' = A$ and $X'' = X$. However, there is no similar element $X^*$ such that for all $A$, $(AX)X^* = A$.

I know there are "right-inverses" and "left-inverses", but is there a name for when the inverse element has to be on the opposite side in order to undo the operation?

Also, I'm referring to inverse elements which isn't strictly accurate according to Wikipedia, but really, these are just elements which reverse multiplication by $X$, but only if on the opposite side of the operation.

Answer 1

If $e$ is an identity element of (S,*) (i.e., S is a unital magma) and $a \times b = e$, then $a$ is called a left inverse of $b$ and $b$ is called a right inverse of $a$. If an element $x$ is both a left inverse and a right inverse of $y$, then $x$ is called a two-sided inverse, or simply an inverse, of $y$.

Answer 2

You should take a look at the commutator.

Depending on the definition that you use, you essentially have that almost all A have a corresponding element X such that the commutator [A,X] is the identity element.

$[A,X] = AXA^{-1}X^{-1}$

(Sorry about formatting I typed this on my phone)