Finally, a field (such as the real numbers) is a commutative ring in which one can do "division" by any nonzero element.

A field is a commutative ring (F,+,*) in which 0 1 and every nonzero element has a multiplicative inverse.

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

First, a nonzero pivot element is selected in a nonbasic column.

A ring in which every nonzero element is a unit and 1 0 is a division ring.

A nonzero nilpotent element is always a zero-divisor (left and right).

Any nonzero element of is contained in only a finite number of height 1 prime ideals.

In a field, every nonzero element is invertible under multiplication; as above, division poses problems only when attempting to divide by zero.

However, in other rings, division by nonzero elements may also pose problems.

The basis functions are usually not orthogonal, so that the overlap matrix S has nonzero nondiagonal elements.