Consider ideals $I_1 = \left(x^{p}\right)$ and $I_2 = \left(y^{q}\right)$, both in $R = \mathbb{R}[x, y, z]$ with the natural grading, $$ R = \bigoplus_{k = 0,1,\dots} R_k\;. $$ Let $R_{\leq k}$ denote the vector space $\sum_{j=0}^k R_k$. Are the following equal?
Question 1: $$ (I_1 + I_2) \cap R_{\leq k} = I_1 \cap R_{\leq k} + I_2 \cap R_{\leq k} $$
Approach
Right hand side is contained in the left hand side. For the other inclusion, $x^p$ and $y^q$ form a Grobner basis and so the following can not occur for polynomials $a_1, a_2 \in R$, $$ \deg(a_1x^p + a_2y^q) < \max(\deg(a_1x^p), \deg(a_2y^q)). $$
Question 2: Will the statement be true if the union of the generators of $I_1$ and $I_2$ did not form a Grobner basis?
If anyone can correct/verify me reasoning. Thank you very much in advance.