Combinatorial properties of the factorial ring (Q1076672)

The paper shows how to exploit a ring (called the factorial ring) in order to give new proofs of classical combinatorial identities such as the recursion formula for Bell numbers: \(B_{n+1}=\sum^{n}_{k=0}\left( \begin{matrix} n\\ k\end{matrix} \right)B_ k\) (recall \(B_ 0=1)\). The factorial ring F is defined as follows: let R be the associative ring with identity over the rationals generated by m commutative and idempotent variables \(x_ 1,x_ 2,...,x_ m\), and let I be the ideal generated by all monomials of degree greater than 1. Then \(F=R/I\).