Left quasi-inverse elements: motivation

Question

An element $a$ in a ring $R$ is a left quasi-inverse if there exists $b\in R$ such that $a+b=ba.$ What is the motivation behind this definition?

Answer 1

Consider an equation like $(1-b)(1-a)=1$, which says that $1-a$ has a left inverse $1-b$. Multiplying through, and cancelling the $1$'s, we are left with $a+b=ba$. This expresses essentially the same identity, but without needing to reference $1$. So, it works perfectly well in nonunital rings, and can thus generalize invertibility to nonunital rings.