Let I and J be ideals of a ring R. Show by example that the set of products {xy | x ∈ I, y ∈ J} need not be an ideal

Question

Let I and J be ideals of a ring R. Show by example that the set of products {xy | x ∈ I, y ∈ J} need not be an ideal, but that the set of finite sums of products of elements of I and J is an ideal. This ideal is called the product ideal, and is denoted by IJ.

I want to check if my work on the first part makes any sense. I am not sure if I am picking the right example.

If I let R = Z and I = (2) and J = (3) then I has multiples of 2, and J has multiples of 3

6 will divide the product IJ if we add 2 and 3 though we get 5 in IJ, but I am not sure if 5 Divides IJ.

Answer 1

Let $I=J=(x,y)\subseteq \mathbb R[x,y]$ notice that $x^2$ and $y^2$ are both in $IJ$ while $x^2+y^2$ is not, because this polynomial is irreducible (so it is no the product of two elements in $I$, because no unit is in $I$).

Answer 2

Jorge's post gives a nice counterexample, but perhaps it is more pedagogically correct to point out that the above set fails to be a new ideal because we have problem with it being a subgroup of the ring $R$. Indeed, being ideal first of all requires being a subgroup of the underlying group structure (abelian) of $R$. By taking the above set only, the sum of two elements will give you for instance an element of the form $xy+zw$, where $x,z \in I$, $y,w \in J$. Though the last might not be written as $xy+zw= \lambda \tau$, for $\lambda \in I$, $\tau \in J$, hence need not be a subgroup necessarily, so neither an ideal!