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 epsilon-closure) appears on the lattice I-A of all ideals of A. For an ideal U of A, the epsilon-closure of U is the largest ideal of A which satisfies the same "multiplicative identities" as U. An algebra A is said to be epsilon-complemented if for every epsilon-closed ideal U of A there exists an epsilon-closed ideal V of A such that A = U circle plus V. What is termed the epsilon'-closure appears in a dual fashion in I-M(A) and the epsilon'-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 pi-closure. We also develop a structure theory for epsilon-complemented algebras. (C) 2011 Elsevier Inc. All rights reserved.