If I define $ I.J=\{ij : i \in I $ & $ j \in J \} $. Then prove that it is not necessrily an ideal, where $I,J$ are ideals in a ring $R$.

Question

If I define $ I.J=\{ij : i \in I $ & $ j \in J \} $. Then prove that it is not necessrily an ideal, where $I,J$ are ideals in a ring $R$.

I have found one counter example in $R[x,y,z]$ for $I=<x,y>$ & $J=<x,z>$ & $R$ is commutative ring without unity.

But can anyone give me any other simple example. Roughly speaking observe what we have show that is $ i_1j_1 + i_2j_2 \neq i_0j_0 $.

Answer 1

Consider the ring $\mathbb Q[x, y]$. Let $I = (x, y)$ and $J = (x + 1, y)$. Then $y = (x + 1)y - xy$ cannot be written as a product of two elements in $I$ and $J$ respectively.