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

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

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

Let G be a group and A a commutative ring.

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

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

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

Connections on a module over a commutative ring always exist.

In general, such functions will form a commutative ring.

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