Abstract algebra | Ring theory

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, or equivalently if the map from R to R that sends x to ax is not injective. 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. 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 is called left regular or left cancellable. Similarly, an element of a ring that is not a right zero divisor is called 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 nonzero ring with no nontrivial zero divisors is called a domain. (Wikipedia).

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Glossary of commutative algebra | Cancellation property | Order (group theory) | Integral domain | If and only if | Finite field | Division ring | Zero ring | Associated prime | Group (mathematics) | Divisibility (ring theory) | Nilpotent | Determinant | Matrix ring | Additive map | Group ring | Pointwise | Element (mathematics) | Unit (ring theory) | Field (mathematics) | Integer | Product of rings | Zero-product property | Noetherian ring | Zero-divisor graph | Endomorphism ring | Prime number | Abstract algebra | Function composition | Domain (ring theory) | Modular arithmetic | Module (mathematics) | Commutative ring