On Jordan superderivations and Jordan super-biderivations of trivial extensions and triangular matrix rings

Triangular matrix rings are example of trivial extensions. In this article we describe the Jordan superderivations of the trivial extensions and upper triangular matrix rings. We deduce then that any Jordan superderivation of an upper triangular matrix ring, under some conditions, is a derivation, and any Jordan super-biderivation of a trivial extension, and hence an upper triangular matrix ring, is a Jordan biderivation.