Let $R$ be a sub-algebra of $\mathbb{Q}[X_1^{\pm 1}, \dots, X_n^{\pm 1}]$ given by finitely many generators, and let $\lambda$ be a linear form $\lambda : \mathbb{Z}^n \to \mathbb{Z}$. This defines a $\mathbb{Z}$-grading on $R$ by setting $\deg(X_1^{v_1} \dots X_n^{v_n}) := \lambda(v_1,\dots,v_n)$.
Are there algorithms for computing generators of the non-negative component $R_+ = \bigoplus_{n \geq 0} R_n$ as a sub-algebra of $\mathbb{Q}[X_1^{\pm 1}, \dots, X_n^{\pm 1}]$?
Thank you in advance!
Added later: Maybe considering arbitrary sub-algebras is too much. I am mainly interested in those sub-algebras which are generated by finitely many monomials (i.e. by elements of the form $X_1^{v_1} \dots X_n^{v_n}$ for some $v_i \in \mathbb{Z}$). Because of this my question can be rephrased in terms of toric rings and toric ideals:
Let $R = \mathbb{Q}[X_1, \dots, X_m]$, $I \trianglelefteq R$ be a toric ideal (i.e. an ideal generated by binomials $X^u - X^v$), and let $\lambda : \mathbb{Z}^n \to \mathbb{Z}$ be a linear form which defines a grading on $R$ as above such that $I$ is homogeneous with respect to $\lambda$.
Is there an algorithm for computing the non-negative component of $R/I$ in terms of toric ideals? To be precise, I want to determine a positive integer $k$ and a toric ideal $J \trianglelefteq S = \mathbb{Q}[Y_1, \dots, Y_k]$ such that $(R/I)_+ \cong S/J$.
