Global dimension of real-exponent polynomial rings (Q6142630)

Let \(k\) be a field. The monoid algebra \(R=R_n=k[\mathbb{R}_+^n] =\{\sum_{a\in\mathbb{R}_+^n}c_ax^a\}\), where \(x^a=x_1^{a_1}\ldots x_n^{a_n}\), is called the ring of real-exponent polynomials in \(n\) variables. The goal of this paper is to show that \(R\) has a global dimension \(n+1\) and flat dimension \(n\). Some ganeralizations are done.