Algebras whose multiplication algebra is semiprime. A decomposition theorem

Any nonassociative algebra A, regarded as a left module over its multiplication algebra M(A), can be endowed with a natural closure: the ε-closure. The ε-closed ideals of A form a complete lattice L(A) with ε-continuous product. The algebra A is said to be ε-decomposable if A is the joint of its atoms in L(A), i.e., A is the ε-closure of the sum of its minimal ε-closed ideals. A distinguished atom is the annihilator Ann(A) of A, whenever it is nonzero. The main result of the paper proves that A is ε-decomposable if, and only if, M(A) is semiprime and any ε-closed ideal U of A, with U≠Ann(A), contains a minimal ε-closed ideal B≠Ann(A). Another characterization of ε-decomposability is also provided, one which involves the notion of ε-radical (the intersection of all maximal ε-closed ideals). This result extends both a Jacobson's theorem for finite-dimensional algebras and a previous one by the authors for algebras with zero annihilator. Moreover, it has well-known precedents (Yood's theorem) in the theory of complete normed algebras.