Jordan-type derivations on trivial extension algebras

Abstract

Assume that [Formula: see text] is a unital algebra over a commutative unital ring [Formula: see text] and [Formula: see text] is an [Formula: see text]-bimodule. A trivial extension algebra [Formula: see text] is defined as an [Formula: see text]-algebra with usual operations of [Formula: see text]-module [Formula: see text] and the multiplication defined by [Formula: see text] for all [Formula: see text] [Formula: see text] In this paper, we prove that under certain conditions every Jordan [Formula: see text]-derivation [Formula: see text] on [Formula: see text] can be expressed as [Formula: see text] where [Formula: see text] is a derivation and [Formula: see text] is both a singular Jordan derivation and an antiderivation. As applications, we characterize Jordan [Formula: see text]-derivations on triangular algebras and generalized matrix algebras.