Zero divisor

In <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, an <a href="/facts/Element_(mathematics)/Q2mx1vHL">element</a> a of a <a href="/facts/Ring_(algebra)/JnCUXz63">ring</a> R is called a left zero divisor if there exists a nonzero x in R such that ax = 0, or equivalently if the <a href="/facts/Function_(mathematics)/CdReEKCh">map</a> from R to R that sends x to ax is not <a href="/facts/Injective/5ES6tqf5">injective</a>. 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 <a href="/facts/Divisibility_(ring_theory)/KQdsMrLw">divisibility in rings</a>. An element that is a left or a right zero divisor is simply called a zero divisor. 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 <a href="/facts/Commutative_ring/QS1r2ovR">commutative</a>, 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, or a non-zero-divisor. A zero divisor that is nonzero is called a nonzero zero divisor or a nontrivial zero divisor. A non-<a href="/facts/Zero_ring/trFjtDJ1">zero</a> ring with no nontrivial zero divisors is called a <a href="/facts/Domain_(ring_theory)/p4eKRC8W">domain</a>.

Zero divisor open-in-new

Zero divisor