Suppose we look at $\mathbb{C}[x_1,\dots,x_n]=\oplus{S_i}$ where $S_i$ are the polynomials of degree $i$. Then we look at a subring $R$ generated by polynomials $p_1,\dots,p_m$ which have some algebraic relations among them. We grade $R=\oplus{R_i}$ where $R_i=R\cap S_i$. Suppose we know an upper bound on the degree that the polynomials $p_i$ can have. Is there an obvious way to transform this into a bound on the Hilbert regularity of $R$? My guess would be that they are the same. Any help would be appreciated.

Suppose we look at the $\mathbb {C}$ module $\mathbb{C}[x_1,\dots,x_n]=\oplus{S_i}$ where $S_i$ are the polynomials of degree $i$. Then we look at a subring $R$ (also a $\mathbb {C} $ module) generated by polynomials $p_1,\dots,p_m$ which have some algebraic relations among them. We grade $R=\oplus{R_i}$ where $R_i=R\cap S_i$. Suppose we know an upper bound on the degree that the polynomials $p_i$ can have. Is there an obvious way to transform this into a bound on the Hilbert regularity of $R$? My guess would be that they are the same. Any help would be appreciated.