Algebras defined by homomorphisms

Let r be a commutative ring with identity, let a and b be two r-algebras and φ: b → a be an r-additive algebra homomorphism. we introduce a new algebra a x φ b, and give some basic properties of this algebra. generalized 2-cocycle derivations on a x φ b are studied. accordingly, a x φ b is considered from the perspective of banach algebras.