Let $R$ be a ring (not necessarily with identity). Is there a name for an element $a\in R$ for which $ab=0$ for all $b\in R$? Similarly, for an element $a\in R$ for which $ba=0$ for all $b\in R$?
Note, a unital ring has exactly one of each, just zero in both cases, as if $ab=0$ for all $b\in R$ then we can take $b=1$.
*Also, examples of non-unital rings that have non-trivial elements of this type would be appreciated.