Let $\mathbb{k}$ be a field of characteristic $0$. Suppose we have a finitely generated graded $\mathbb{k}$-algebra $A= \bigoplus_{i=0}^{\infty}A_i$ which is free of finite rank as a module over a graded ring $R= \bigoplus_{i=0}^{\infty}R_i$, where $R_0=\mathbb{k}$. Let $B=\{m_1,...,m_n\}$ be an $R$-basis for $A$, and let $d=\max\{\deg(m_i) : i \leq n \}$.
Define the structure constants $c_{i,j}^k \in R$ to be the ring elements that appear in the sum $$m_im_j=\sum_k c_{i,j}^km_k.$$ For what algebras $A$ is it the case that for all $i,j \leq n$ such that $i+j \leq d$, there exists some $k \leq n$ such that $c_{i,j}^k \in \mathbb{k}$? Are there necessary and sufficient conditions one can state for this to be the case?