Let $R = \bigoplus_{n \in \mathbb{Z}} R_n$ be a $\mathbb{Z}$-graded commutative ring with nonnegative part $R^+ = \bigoplus_{n \geq 0} R_n$ and nonpositive part $R^- = \bigoplus_{n \geq 0} R_{-n}$. By a well known theorem the following statements are equivalent:

- $R$ is Noetherian
- $R_0$ is Noetherian and both $R^+$ and $R^-$ are finitely generated $R_0$-algebras

In particular, the second item holds if $R$ is a finitely generated algebra over some field $k$.

**My Question:** Are there algorithms for computing $R^+$ in terms of finitely many generators for some special finitely generated $k$-algebras $R$?

I am mainly interested in the case where $R$ is a toric ring and the grading is given by a linear form $\lambda : \mathbb{Z}^n \to \mathbb{Z}$. To be specific, $R$ is a $k$-subalgebra of the ring of Laurent-polynomials $k[X_1^{\pm 1}, \dots, X_n^{\pm n}]$ generated by finitely many monomials $X^v = X_1^{v_1} \dots X_n^{v_n}$ for some vectors $v \in \mathbb{Z}^n$. A monomial $X^v$ is by definition of degree $\deg(X^v) = \lambda(v)$.

Geometrically the monomials of $R$ correspond to the integral points of a rational polyhedral cone in $\mathbb{R}^n$, and computing the nonnegative part with respect to $\lambda$ corresponds to taking the intersection of this cone with the rational half-space given by $\lambda$.