Rings, Semiprime Ideals

Semiprime Ideals

An ideal C in R is semiprime if, for any ideal A, A² in C implies A is in C. Note that C can equal R. A prime ideal must be proper, but a semiprime ideal need not be. Also, every prime ideal is semiprime.

Replace A² with Aⁿ, and get an equivalent definition. If Aⁿ lies in C then so does A^n/2 and A^n/4, and so on down to A.

The xRy test for prime ideals has a counterpart for semiprime ideals. Review the earlier proof, and apply it to A*A instead of A*B. The ideal C is semiprime iff xRx in C implies x is in C, for all x in R.

The homomorphic image of a semiprime ideal is semiprime. Apply the proof for prime ideals.

The intersection of semiprime ideals is semiprime; A² in all of them implies A is in all of them. This is new; the intersection of prime ideals need not be prime.

The intersection of a descending chain of semiprime ideals is semiprime, hence every semiprime ideal contains a minimal semiprime ideal. Apply the proof for prime ideals. Once again the minimal semiprime ideal beneath a given semiprime ideal is always 0 in a domain.