In a previous paper [J. Gubeladze, Invent. Math. 160, No. 1, 173-216 (2005; Zbl 1075.14051)] the author showed that for any additive submonoid \(M\) of a rational vector space with trivial group of units one has a nilpotent action of the multiplicative monoid \({\mathbb N}\) on the quotient group \(K_{i}( k[M])/K_{i}(k).\) This means that for any sequence of natural numbers \(c_{1},c_{2},\dots \geq 2\) and \(x\in K_{i}( k[M])\) one has \((c_{1}\cdots c_{j})_{*}(x)\in K_{i}(k)\) for all \(j\gg 0\) (\(j\) potentially depends on \(x\)). \(k\) was assumed to be a field of characteristic zero. In the current paper the author shows the following theorem: Let \(M\) be an additive submonoid of a \({\mathbb Q}\)-vector space with a trivial group of units. Then for any regular ring \(R\) with \({\mathbb Q}\subset R\) the multiplicative monoid \({\mathbb N}\) acts nilpotently on \(K_{i}(R[M])/K_{i}(R), \,\, i\geq 0\).