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.

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