Zero divisor
Ring element that can be multiplied by a non-zero element to equal 0 / From Wikipedia, the free encyclopedia
Dear Wikiwand AI, let's keep it short by simply answering these key questions:
Can you list the top facts and stats about Zero divisor?
Summarize this article for a 10 year old
In abstract algebra, an element a of a ring R is called a left zero divisor if there exists a nonzero x in R such that ax = 0,[1] or equivalently if the map from R to R that sends x to ax is not injective.[lower-alpha 1] Similarly, an element a of a ring is called a right zero divisor if there exists a nonzero y in R such that ya = 0. This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor.[2] An element a that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero x such that ax = 0 may be different from the nonzero y such that ya = 0). If the ring is commutative, then the left and right zero divisors are the same.
An element of a ring that is not a left zero divisor (respectively, not a right zero divisor) is called left regular or left cancellable (respectively, right regular or right cancellable). An element of a ring that is left and right cancellable, and is hence not a zero divisor, is called regular or cancellable,[3] or a non-zero-divisor. A zero divisor that is nonzero is called a nonzero zero divisor or a nontrivial zero divisor. A non-zero ring with no nontrivial zero divisors is called a domain.