If R is a commutative ring, then the above three notions are all the same.

A commutative ring where the set with multiplication is a group.

For general commutative rings, we don't have a field of fractions.

In general, such functions will form a commutative ring.

Let G be a group and A a commutative ring.

Connections on a module over a commutative ring always exist.

For commutative rings, all three concepts coincide, but in general they are different.

For commutative rings this definition is equivalent to the one given in the previous section.

Let R be a commutative ring with identity 1.

The rational, real and complex numbers form commutative rings (in fact, they are even fields).

This is the standard definition of the resultant over a commutative ring.

Therefore, by definition, any field is a commutative ring.

X is homeomorphic to the spectrum of a commutative ring.

Integral domains and fields are especially important types of commutative rings.

The two definitions can be different for commutative rings which are not Noetherian.

For simplicity, we will consider only the case of simplicial commutative rings.

When A is not a commutative ring, the idea of order is still important, but the phenomena are different.

The following conditions on a commutative ring R are equivalent:

This implies it is a nontrivial commutative ring with identity.

The real numbers can be extended to a wheel, as can any commutative ring.

In general, two elements in a commutative ring can have no least common multiple or more than one.

A Mori domain in mathematics is a type of commutative ring.

A commutative ring is always equal to its opposite ring.

An example of a commutative ring is the set of integers.

For commutative rings the left and right definitions coincide, but in general they are distinct from each other.