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In a commutative ring R, the set of non-zero-divisors is a multiplicative set in R. (This, in turn, is important for the definition of the total quotient ring.) The same is true of the set of non-left-zero-divisors and the set of non-right-zero-divisors in an arbitrary ring, commutative or not.
If one interprets the definition of divisor literally, every a is a divisor of 0, since one can take x = 0. Because of this, it is traditional to abuse terminology by making an exception for zero divisors: one calls an element a in a commutative ring a zero divisor if there exists a nonzero x such that ax = 0. [2]
An integral domain is a nonzero commutative ring with no nonzero zero divisors. An integral domain is a commutative ring in which the zero ideal {0} is a prime ideal. An integral domain is a nonzero commutative ring for which every nonzero element is cancellable under multiplication.
The zero ring is the unique ring of characteristic 1. The only module for the zero ring is the zero module. It is free of rank א for any cardinal number א. The zero ring is not a local ring. It is, however, a semilocal ring. The zero ring is Artinian and (therefore) Noetherian. The spectrum of the zero ring is the empty scheme. [8]
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.
Zero divisors have a topological interpretation, at least in the case of commutative rings: a ring R is an integral domain if and only if it is reduced and its spectrum Spec R is an irreducible topological space. The first property is often considered to encode some infinitesimal information, whereas the second one is more geometric.
More generally, every integral domain is a reduced ring since a nilpotent element is a fortiori a zero-divisor. On the other hand, not every reduced ring is an integral domain; for example, the ring Z[x, y]/(xy) contains x + (xy) and y + (xy) as zero-divisors, but no non-zero nilpotent elements.
In an integral domain (a commutative ring without zero divisors), every non-zero element is regular, so a torsion element of a module over an integral domain is one annihilated by a non-zero element of the integral domain. Some authors use this as the definition of a torsion element, but this definition does not work well over more general rings.