Generalized \(d\)-derivations of rings without unit elements (Q2758092)

Let \(A\) be an algebra over a commutative unital ring and let \(M\) be a \(K\)-bimodule. Let \(d\colon A\to M\) be a derivation and let \(f\colon A\to M\) be a linear map satisfying \(f(ab)=f(a)b+ad(b)\) for all \(a,b\in A\). The author calls the pair \((f,d)\) of such maps a Brešar generalized derivation (the study of such pairs was initiated by the reviewer [in Glasg. Math. J. 33, No. 1, 89-93 (1991; Zbl 0731.47037)]). Further, if \(f\colon A\to M\) is a linear map such that for some \(m\in M\) we have \(f(ab)=f(a)b+af(b)+amb\) for all \(a,b\in A\), then the pair \((f,m)\) is called a generalized derivation. This concept was introduced by \textit{A. Nakayima} [in Sci. Math. 2, No. 3, 345-352 (1999; Zbl 0968.16018)]. One of the main results of the paper characterizes when the notions of Brešar generalized derivations and generalized derivations essentially coincide. The problem of extending Brešar generalized derivations from \(A\) to the unitization of \(A\) is discussed, and the so-called (Brešar) generalized Jordan derivations are treated.