Absorbing element

In algebra, an absorbing element or a zero element for a binary operation has a property similar to that of multiplication by zero.

Formally, let $$\star$$ be a binary operation on a set X. An element O of X is absorbing for $$\star$$ if


 * $$O \star x = O = x \star O \,$$

holds for all x in X. An absorbing element, if it exists, is unique.

Examples

 * The zero (additive identity element) of a ring is an absorbing element for the ring multiplication.
 * The zero matrix is the absorbing element for matrix multiplication.