Product of two ideals in polynomial rings

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It is easy I want to make sure if got the idea right

Assume $R=Z[x,y]$, $I=<xy,y^2>$, $J=<x^2,xy>$ are two ideals of $R$ then

$ I J=<x^3 y,x^2 y^2,x^2 y^2, x y^3>$. Is it right?

Any suggestions will be appreciated