Suppose we are given two polynomial rings $R_1$ and $R_2$ by presenting their generators, $S_1$ and $S_2$, where $S_i$ are finite set of $m_i$ variables, $i.e.$, $S_i\subset P[x_1,x_2,\cdots,x_{m_i}]$. $R_i$ are generated from $S_i$ by the usual addition, subtraction, and multiplication.

The goal is to compute the intersection ring of $R_1$ and $R_2$ by given some generating set of $R_1\cap R_2$. Is $R_1\cap R_2$ still finitely generated?

How about general case that there are $n$ rings, and $S_i$ only consists of homogeneous polynomials with degree $l_i$, can we bound the degree of some generating set of $\cap_{i=1}^n R_i$?