Higher \(R\)-derivations of special subrings of matrix rings (Q1061820)

This is one of a series of papers by the author on higher order derivations on special subrings of matrix rings. Recall a sequence of mappings \(d_ m: R\to R\) is called a derivation of order \(S\) if (1) \(m\in S\), a sequence of natural numbers, (2) \(d_ m(a+b)=d_ ma+d_ mb\), (3) \(d_ m(ab)=\sum d_ i(a)d_ j(b)\) where the sum is over \(i\) and \(j\) such that \(i+j=m\), and (4) \(d_ 0(a)=a\), for all \(m\in S\) and \(a, b\in R\). A special subring of a matrix ring is a subring whose \((i,j)\) entry is \(0\) when \((i,j)\) is in a special relation which is reflexive and transitive. The notion inner derivations of order \(S\) is similarly defined. A new notion, transitive mapping, is defined to reflect the special relation mentioned above. The following results are proved: necessary and sufficient conditions for such derivations to be inner; a representation of such a derivation by inner derivations; finally the notion of integrable derivation is introduced with a detailed examination for the conditions under which such derivations are integrable. This paper is of interest to people interested in derivations and their generalizations.