Prime ideal

A (two-sided) proper ideal is called a prime ideal iff for any , implies or ¹, #m/def/ring i.e.

Historical note

Considering the original notion of an Ideal number, an ideal is the set of multiples of an ideal number . Therefore the above is equivalent to

i.e. is prime.

Note an ideal is prime iff where is prime or zero. The set of all prime ideals of a commutative ring is called its spectrum.

Properties