Replace A2 with An, and get an equivalent definition. If An lies in C then so does An/2 and An/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.