# Atomicity of the semidomain (under Dirichlet convolution) of arith. fncs to a subrig of the ring of integers of a totally real NF

Let $S$ be a non-trivial subsemiring of the ring of integers of a totally real number field, and denote by $D_S$ the integral semidomain of all arithmetic functions $\mathbf N^+ \to S$ with the operations of pointwise addition and Dirichlet convolution. It is clear that $D_S$ is atomic (respectively, factorial) only if so is $S$, and my questions are somehow on the converse of this statement:

Q1. Is there any case at all in which $D_S$ is atomic (respectively, factorial)? Is there anything in the literature about that?

I'm especially interested in the case when $S$ is a ring, and don't even know the answer for $S = \mathbf Z$.

Let me recall that an integral semidomain is atomic if every non-zero, non-unit element is a (finite) product of atoms (irreducible elements) in at least one way, and factorial if every non-zero, non-unit element can be expressed as a product of atoms in an essentially unique way.

Q2. What about the atomicity of the subrig of $D_S$ consisting of finitely supported arithmetic functions?

• I see I got a negative vote (counterbalanced by a subsequent positive one). It is hard to divine why without a comment by the voter, but I've tried to guess and improve the post. – Salvo Tringali Feb 5 '17 at 8:42
• Regarding Q1 and factoriality, see this 1963 paper of Cashwell and Everett: projecteuclid.org/euclid.pjm/1103035955 – so-called friend Don Feb 5 '17 at 19:42