On formal integration in composition rings of polynomials (Q1177099)

The formal integration on polynomials was introduced by \textit{W. B. Müller} in several pioneering papers [see Math. Slovaca 33, 121-126 (1983; Zbl 0512.12015)]. In this paper the composition ring \(A[x]\) of all polynomials over a quotient ring \(A\) is considered. The formal integration is a mapping \(I:A[x]\to A[x]\) with four properties. Theorem 1 gives a complete description: all formal integrations are given by \(I(a_ 0+a_ 1x+\cdots+a_ nx_ n)=\lambda a_ 0x+{\lambda a_ 1x^ 2\over\lambda+1}+\cdots{\lambda a_ nx^{n+1}\over\lambda n+1}\) where \(\lambda\) is an idempotent of \(A\) such that \((1-\lambda)A\) contains all elements of finite order of \(A\). Another beautiful result connects the formal derivation to the formal integration. For the formal derivation \(D=\lambda{d\over dx}\) with \(\lambda\) described as above it follows that \(D(I(f))=\lambda f\) for all \(f\in A[X]\). \(D\circ I\) is the identical map on \(\lambda A[x]\). Conversely \(I(D(f))=\lambda(f-a_ 0)\) where \(a_ 0\) is the constant term of \(f\).