A prime ideal is an ideal I such that if a b element I, then either a element I or b element I. For example, in the integers, the ideal a = 〈p〉 (i.e., the multiples of p) is prime whenever p is a prime number. In any principal ideal domain, prime ideals are generated by prime elements. Prime ideals generalize the concept of primality to more general commutative rings. An ideal I is prime iff the quotient ring R/I is an integral domain because x element I iff x congruent 0 (mod I). Technically, some authors choose not to allow the trivial ring R = {0} as a commutative ring, in which case they usually require prime ideals to be proper ideals.
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