Summary: Let \(R\) be a commutative Noetherian ring of dimension \(d\), \(M\) a commutative cancellative torsion-free monoid of rank \(r\) and \(P\) a finitely generated projective \(R[M]\)-module of rank \(t\). Assume \(M\) is \(\Phi\)-simplicial seminormal. If \(M\in \mathcal {C}({\Phi})\), then Serre \(\dim R[M]\leq d\). If \(r\), then Serre \(\dim R[\mathrm{int}(M)]\leq d\). If \(M\subset \mathbb {Z}_{+}^{2}\) is a normal monoid of rank 2, then Serre \(\dim R[M]\leq d\). Assume \(M\) is \(c\)-divisible, \(d=1\) and \(t\). Then \(P \cong\wedge^t P\otimes R[M]^{t-1}\). Assume \(R\) is a uni-branched affine algebra over an algebraically closed field and \(d=1\). Then \(P \cong\wedge^t P\otimes R[M]^{t-1}\).