In this paper we investigate some relations among a local (non-associative) ring $S$ and its subsets $L$, $L_d$, $S\backslash L$ and $A$, and their associators in $S$ (where $L(L_d)$ is the set of all elements from $S$ which do not have a left (right) inverse in $S$ and $A$ the set of all elements which do not have neither left nor right inverses).