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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]
First, this agrees with the definition that a domain is a ring in which 0 is the only zero divisor (in particular, 0 is required to be a zero divisor, which fails in the zero ring). Second, this way, for a positive integer n , the ring Z / n Z is a domain if and only if n is prime, but 1 is not prime.
No nilpotent element can be a unit (except in the trivial ring, which has only a single element 0 = 1).All nilpotent elements are zero divisors.. An matrix with entries from a field is nilpotent if and only if its characteristic polynomial is .
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.
The ring of integers Z is a reduced ring. Every field and every polynomial ring over a field (in arbitrarily many variables) is a reduced ring. More generally, every integral domain is a reduced ring since a nilpotent element is a fortiori a zero-divisor.
In a commutative Artinian ring, every maximal ideal is a minimal prime ideal.; In an integral domain, the only minimal prime ideal is the zero ideal.; In the ring Z of integers, the minimal prime ideals over a nonzero principal ideal (n) are the principal ideals (p), where p is a prime divisor of n.
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.