The set of all elements in such that . Here is a ring or a semi-group (or, generally, a groupoid) with a zero. The right annihilator of a set in is defined in a similar manner as the set The set is the two-sided annihilator of . In an associative ring (or semi-group) the left annihilator of an arbitrary set is a left ideal, and if is a left ideal of , then is a two-sided ideal of ; in the non-associative case these statements are usually not true.