Nagata's principle of idealization in relation to module homomorphisms and conditional expectations. (Q2718046)

Let \(R\) be a commutative ring and \(M\) be an \(R\)-module. Then \(R\times M\) equipped with the actions \((r,a)+(s,b)=(r+s, a+b)\) and \((r,a)(s,b)=(rs, rb+sa)\) is a ring containing the copy \(\{0\}\times M\) of \(M\). This ring is denoted by \(R(+)M\) and called Nagata's principle of idealization of \(M\) over \(R\) [cf. \textit{M. Nagata} in: Proc. Internat. Sympos. algebraic number theory, Tokyo \& Nikko 1955, 191--226 (1956; Zbl 0075.02301)].NEWLINENEWLINEThe authors describe the structure of \(R(+)M\) as the ring of all triangular matrices of the form \(\left ( \begin{smallmatrix} r&0 \\ a&r\end{smallmatrix} \right )\) in a natural way. They also show that if \(\phi\) and \(\phi^{\prime}\) are unital \(R\)-module homomorphisms from \(S\) to \(R\) where \(R\) and \(S\) are assumed to be rings with \(R\subseteq S\) and with common identity, then \(R(+)\ker\phi\) and \(R(+)\ker\phi^{\prime}\) are isomorphic rings.NEWLINENEWLINESome application of the obtained results to certain Banach algebras and rings of operators acting on the Hilbert space \({\mathcal L}^{2}(X,{\mathcal A}, m)\) where \((X,{\mathcal A}, m)\) is a complete probability space are given. In particular, it is shown that if \(W\) is an abelian von Neumann algebra of operators acting on a separable Hilbert space \(H\) then there is an algebra \(N\) of nilpotent operators of index \(2\) such that \(W(+)N\) is a maximal abelian algebra of operators on \(H\).