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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.
The naive solution is to replace the fraction field by the total quotient ring, that is, to invert every element that is not a zero divisor. Unfortunately, in general, the total quotient ring does not produce a presheaf much less a sheaf. The well-known article of Kleiman, listed in the bibliography, gives such an example.
Let K be a field, and G a torsion-free group. Kaplansky's zero divisor conjecture states: The group ring K[G] does not contain nontrivial zero divisors, that is, it is a domain. Two related conjectures are known as, respectively, Kaplansky's idempotent conjecture: K[G] does not contain any non-trivial idempotents, i.e., if a 2 = a, then a = 1 ...
An example: the ring k[x, y]/(xy), where k is a field, is not a domain, since the images of x and y in this ring are zero divisors. Geometrically, this corresponds to the fact that the spectrum of this ring, which is the union of the lines x = 0 and y = 0 , is not irreducible.
But in the ring Z/6Z, 2 is a zero divisor. This equation has two distinct solutions, x = 1 and x = 4, so the expression is undefined. In field theory, the expression is only shorthand for the formal expression ab −1, where b −1 is the multiplicative inverse of b.
As another example, the ring Z × Z contains (1, 0) and (0, 1) as zero-divisors, but contains no non-zero nilpotent elements. The ring Z/6Z is reduced, however Z/4Z is not reduced: the class 2 + 4Z is nilpotent. In general, Z/nZ is reduced if and only if n = 0 or n is square-free. If R is a commutative ring and N is its nilradical, then the ...
If K is a field, then the ring K[x,y,z]/(xy,xz) (the coordinate ring of the union of a line and a plane) is reduced, but not equidimensional, and hence not Cohen–Macaulay. Taking the quotient by the non-zero-divisor x−z gives the previous example.
In finite field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite field GF(p m).This means that a polynomial F(X) of degree m with coefficients in GF(p) = Z/pZ is a primitive polynomial if it is monic and has a root α in GF(p m) such that {,,,,, …} is the entire field GF(p m).