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.
• The empty set is the absorbing element for intersection of sets.