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A divisor of that is not a trivial divisor is known as a non-trivial divisor (or strict divisor [6]). A nonzero integer with at least one non-trivial divisor is known as a composite number, while the units −1 and 1 and prime numbers have no non-trivial divisors. There are divisibility rules that allow one to recognize certain divisors of a ...
The ring of integers modulo a prime number has no nonzero zero divisors. Since every nonzero element is a unit, this ring is a finite field. More generally, a division ring has no nonzero zero divisors. A non-zero commutative ring whose only zero divisor is 0 is called an integral domain.
A divisor on Spec Z is a formal sum of prime numbers with integer coefficients and therefore corresponds to a non-zero fractional ideal in Q. A similar characterization is true for divisors on Spec O K , {\displaystyle \operatorname {Spec} {\mathcal {O}}_{K},} where K is a number field.
The greatest common divisor (GCD) of integers a and b, at least one of which is nonzero, is the greatest positive integer d such that d is a divisor of both a and b; that is, there are integers e and f such that a = de and b = df, and d is the largest such integer.
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]
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.
In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, =, = = This property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nontrivial zero divisors, or one of the two zero-factor properties. [1]
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.