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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.
When R is commutative, the notions of left divisor, right divisor, and two-sided divisor coincide, so one says simply that a is a divisor of b, or that b is a multiple of a, and one writes . Elements a and b of an integral domain are associates if both a ∣ b {\displaystyle a\mid b} and b ∣ a {\displaystyle b\mid a} .
A left zero divisor of a ring R is an element a in the ring such that there exists a nonzero element b of R such that ab = 0. [d] A right zero divisor is defined similarly. A nilpotent element is an element a such that a n = 0 for some n > 0. One example of a nilpotent element is a nilpotent matrix.
An element r of R is a called a two-sided zero divisor if it is both a left zero divisor and a right zero divisor. division A division ring or skew field is a ring in which every nonzero element is a unit and 1 ≠ 0. domain A domain is a nonzero ring with no zero divisors except 0.
(Here we allow zero to be a zero divisor.) In particular D R is the set of (left) zero divisors of R taking S = R and R acting on itself as a left R -module. When R is commutative and Noetherian, the set D R {\displaystyle D_{R}} is precisely equal to the union of the associated primes of the R -module R .
In particular, any quasinilpotent element is a topological divisor of zero (e.g. the Volterra operator). An operator on a Banach space X {\displaystyle X} , which is injective , not surjective , but whose image is dense in X {\displaystyle X} , is a left topological divisor of zero.
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In algebra, a domain is a nonzero ring in which ab = 0 implies a = 0 or b = 0. [1] ( Sometimes such a ring is said to "have the zero-product property".) Equivalently, a domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor).