# Catenarity of monoid algebras

Let $R$ be a commutative ring, let $M$ be a commutative monoid, and let $R[M]$ denote the corresponding monoid algebra. Suppose further that $R$ is universally catenary. One may ask for conditions on $M$ such that the ring $R[M]$ is catenary. I know of the following results:

1. If $M$ is of finite type then $R[M]$ is catenary. (Trivial.)

2. If $M$ is cancellative and torsionfree of rank $1$, then $R[M]$ is catenary. (Follows from Corollary 2.15 in S.Améziane, D.E.Dobbs, S.Kabbaj, On the prime spectrum of commutative semigroup rings, Comm. Alg. 26 (1998), 2559-2589. The noetherian hypothesis is not used in the part relevant to this question.)

3. If $M$ is a torsionfree group of finite rank, then $R[M]$ is catenary. (Follows from Theorem 3.3 in S. Améziane, D. Costa, S. Kabbaj, S. Zarzuela, On the spectrum of the group ring, Comm. Alg. 27 (1999), 387-403.

Without hypothesis on $M$ the ring $R[M]$ need not be catenary. For example, the polynomial ring over $\mathbb{Q}$ in countably infinitely many indeterminates is not catenary by Example 2.15 in D.E.Dobbs, M.Fontana, S.Kabbaj, Direct limits of Jaffard domains and S-domains, Comment. Math. Univ. St. Pauli 39 (1990).

Are there further results in this direction, e.g., is it known whether $R[M]$ is catenary if $M$ is cancellative and torsionfree of finite rank?