More generally, there are no zero divisors in division rings.

A biquaternion is either a unit or a zero divisor.

If the ring is commutative, then the left and right zero divisors are the same.

However, the ring constructed in this way contains zero divisors and thus cannot be a field.

That is, it is a ring which has no left or right zero divisors.

In an Artinian ring, all elements are units or zero divisors.

Some authors opt to include zero as a zero divisor.

Clearly, if then q is a zero divisor.

Here is an example of a ring with an element that is a zero divisor on one side only.

A commutative ring without zero divisors and in which 1 0 is an integral domain.

Nothing more in the small ring can be given an inverse, because zero divisors are impossible to invert.

If a non-trivial ring R does not have any zero divisors, then its characteristic is either 0 or prime.

In an Artinian ring, each element is either invertible or a zero divisor.

Then it is possible to have zero divisors in the ring of regular functions, and consequently the fraction field no longer exists.

In fact, all of the nonzero imaginary elements are zero divisors (also see the section "Division").

They contain non-trivial idempotents and zero divisors, but no nilpotents.

The matrix ring of order greater than one is never a domain, since it has zero divisors, and even nilpotent elements.

Elements with multiplicative inverses can still be zero divisors.

A non-zero element of a ring that is not a zero divisor is called regular.

He also showed how zero divisors arise in tessarines, inspiring him to use the term "impossibles."

The set of zero divisors is the union of the associated prime ideals of the ring.

They all have zero divisors.

Their modulus is defined differently from their norm, and they also contain zero divisors.

If the ring or algebra is finite, however, then all elements a which are not zero divisors do have a (left and right) inverse.

However, choosing a reducible polynomial can result in missed errors, due to the rings having zero divisors.