ε-Complemented algebras

For a nonassociative algebra A, by considering A as a left module over its multiplication algebra M(A), a closure operation (termed the ε-closure) appears on the lattice IA of all ideals of A. For an ideal U of A, the ε-closure of U is the largest ideal of A which satisfies the same “multiplicative identities” as U. An algebra A is said to be ε-complemented if for every ε-closed ideal U of A there exists an ε-closed ideal V of A such that A=U⊕V. What is termed the ε′-closure appears in a dual fashion in IM(A) and the ε′-complementarity can be considered in M(A). This paper provides different characterizations of both complementarities. Moreover, we determine the relation between these concepts, the classical complementarity, and the complementarity for the π-closure. We also develop a structure theory for ε-complemented algebras.