If an algebra is graded by the group $G$: $A=\bigoplus\limits_{d \in G} A_d$ and contains a homogeneous ideal $I \subset A$, then we have the quotient $B:=A/I$ and canonical epimorphism $\nu:A \rightarrow B$. Now $B$ is $K$-graded by $B=\bigoplus\limits_{d \in K} B_d$ where $B_d:=A_d+I$.
Suppose now we know only the result of factorization, an algebra $B$ and $K$-homogeneous generators $f_1, \dots, f_s$ of an ideal $I$ of $A$. We would like to recover the algebra we start with from that data. Say, to know it as a quotient of a polynomial algebra by an ideal of relations. Is there a way to find a unique (up to isomorphism) finitely generated algebra $A$ such that $A/I=B$?