Genus Characters and the Arithmetic of Commutative Monoids Original Research Article

In this paper we develop an ideal theory for certain submonoids of the nonzero integers. We associate one of these monoids to each quadratic number field and show that the genus theory of ideals and genus characters of the number field are virtually the same as the ideal theory and the characters of the ideal class group of the monoid. We define L-functions associated to characters of the class group of a finite intersection of these monoids and prove an analogue of Dirichletʹs theorem on primes in an arithmetic progression. We further prove a curious relationship between the nature of the singularity of the zeta function and the 2-rank of the class group.