A derimorphism over a ring is a mixture of a derivation and a
homomorphism. A level \(\lambda=(\lambda_1, \lambda_2, \lambda_3)\)
derimorphism \(D\) over a ring \(R\) is an additive mapping over
\(R\) such that \(D(xy)=D(x)y+xD(y)-\lambda D(x)D(y)\) for some
central element \(\lambda\in Cen(R)^3\), where \(Cen(R)\) is the
center of \(R\). The usual derivation is just a derimorphism of
level \((1, 1, 0)\) while the backward (respectively forward)
\(h\)-difference a derimorphism of level (1, 1, 1) (respectively
\((1, 1, -1)\)). A general theory of derimorphisms over a ring with
identity is developed in this paper, in particular, a
Singer-Wermer-Thomas type theorem, that is, the range of a
derivation over a commutative Banach algebra is contained in the
radical, is proved for elementary algebras (possibly infinite
dimensional).
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