zero divisor

Noun

 * 1)  An element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.
 * 2)  A nonzero element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.
 * 1)  A nonzero element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.
 * 1)  A nonzero element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.
 * 1)  A nonzero element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.
 * 1)  A nonzero element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.

Usage notes

 * The two definitions differ according to whether or not 0 is considered a zero divisor.
 * The definition that includes 0 is the one preferred by . (See reference cited in )
 * Additionally, 0 may be called the.
 * Related terminology:
 * An element (resp. nonzero element) $$a$$ such that $$\exists x\in R:x \ne 0 \and ax=0$$ is called a.
 * An element (resp. nonzero element) $$a$$ such that $$\exists x\in R:x \ne 0 \and xa=0$$ is called a.
 * An element $$a$$ that is both a left zero divisor and a right zero divisor is called a.
 * Thus, a zero divisor can be (and often is) defined as any element that is either a left zero divisor or a right zero divisor.
 * The term zero divisor is most relevant in the context of commutative rings (where the left-right distinction is not made).

Translations

 * Finnish: nollanjakaja
 * French: