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 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?

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Source : Link , Question Author : Fred Rohrer , Answer Author : Community

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