Zero divisor

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.^[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 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.

^ N. Bourbaki (1989), Algebra I, Chapters 1–3, Springer-Verlag, p. 98
^ Charles Lanski (2005), Concepts in Abstract Algebra, American Mathematical Soc., p. 342
^ Nicolas Bourbaki (1998). Algebra I. Springer Science+Business Media. p. 15.

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[1] N. Bourbaki (1989), Algebra I, Chapters 1–3, Springer-Verlag, p. 98

[3] Charles Lanski (2005), Concepts in Abstract Algebra, American Mathematical Soc., p. 342

[4] Nicolas Bourbaki (1998). Algebra I. Springer Science+Business Media. p. 15.

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