reduced ring

Definition

A commutative ring is called reduced if for an element , then for any integer .

Said another way, it is reduced if it has no non-zero nil-potent elements.

Connection to quotient and radical ideals

Let be a reduced ring and an ideal. Then is reduced if and only if is radical.

Proof

Assume is reduced. Then is nilpotent of order if and only if and is non-zero if and only if .

Using the nilradical

A ring is reduced if and only if the nilradical .

We can use this to “make” the ring reduced

Connection to prime spectrum

There is a canonical projection map

Using the prime spectrum functor, we have

which gives on the right hand side the maximal subspace on which the functions are “geometric” in some sense. Note the left hand side corresponds with affine varieties where the right hand side corresponds with schemes.