## The product of homogeneous polynomials is homogeneous

Prove that the product of homogeneous polynomials is homogeneous.

Let $R$ be a ring and let $p,q \in R[x_1,\ldots,x_n]$ be homogeneous of degree $t$ and $u$ respectively; say $p = \sum_{i=1}^a p_i \prod_{j=1}^n x_j^{t_{i,j}}$ and $q = \sum_{k=1}^b q_k \prod_{j=1}^n x_j^{u_{k,j}}$ with $\sum_j t_{i,j} = t$ and $\sum_j u_{k,j} = u$ for all $i$ and $k$, respectively.

Then $pq = \sum_{i=1}^a \sum_{k=1}^b p_iq_k \prod_{j=1}^n x_j^{t_{i,j} + u_{k,j}}$. Note that for each $(i,k)$, we have $\sum_j t_{i,j} + u_{k,j} = t+u$. Thus $pq$ is homogeneous.