Structure theory for multiplicatively semiprime algebras

In this paper we shall develop a structure theory for multiplicatively semiprime algebras. To this end we shall introduce an algebraic closure for ideals of an algebra A, which involves the multiplication algebra M(A) of A. This closure is called the -closure, and is stronger than the classic closure (referred to as the π-closure). An algebra A is said to be multiplicatively semiprime (in short m.s.p.) (respectively multiplicatively prime (in short m.p.)) whenever both A and M(A) are semiprime (respectively prime) algebras. We will prove that m.s.p. algebras are just semiprime algebras which satisfy the following equivalent assertions: (i) Both and π closures agree. (ii) The lattice of all -closed ideals is an annihilator lattice. We will establish a Yood theorem for algebras with zero annihilator and also a description theorem for atomic m.s.p. algebras. Summarizing these results: For an algebra A with zero annihilator, the following assertions are equivalent: (i) A is -decomposable. (ii) A is an atomic m.s.p. algebra. (iii) A is an essential subdirect product of a family of m.p. algebras. These results are also applied to normed algebras, and especially to generalized annihilator normed algebras.