Closed prime ideals in algebras with semiprime multiplication algebra

Let A be an algebra whose multiplication algebra M(A) is semiprime. We prove that, except in an exceptional case, the proper closed prime ideals of A are the maximal closed ideals of A, for the closure operations pi and epsilon. lit fact, these sets agree for both closures. The same can he said in M(A) for the closure operations pi and epsilon'. Moreover, we establish the relationships between the proper closed prime ideals of A and the ones of the algebras M(A), U and A/U, for a given ideal U of A.