On endomorphisms of degree two
Let \(R\) be a commutative ring and \(\Delta\in R\). If \(A\) is an \(R\)-module and \(f\colon A\to A\) is an endomorphism satisfying \(f^2=\Delta \cdot \text{id}_A\), the pair \((A,f)\) is said to be a \(\Delta\)-pair. A homomorphism \(\phi\colon (A,f)\to (B,g)\) of \(\Delta\)-pairs is a linear map \(\phi\colon A\to B\) such that \(\phi f = g\phi\). The author observes that, for two \(\Delta\)-pairs \((A,f)\) and \((B,g)\), the submodule of \(A\otimes B\) spanned by the elements \(f(a)\otimes b+a\otimes g(b)\), where \(a\in A\), \(b\in B\), is again a \(\Delta\)-pair. This gives rise to a composition of isomorphism classes of \(\Delta\)-pairs which is commutative and associative. In the case \(R = \mathbb Z\) and the modules are free of rank 2, a suitably chosen subset of isomorphism classes of \(\Delta\)-pairs corresponds bijectively to ideal classes in the ring \(\mathbb Z[\sqrt{\Delta}] \). Thus the composition of \(\Delta\)-pairs generalizes the classical composition of binary quadratic forms.
scientific article; zbMATH DE number 4077328