What would be an example such that $\langle a\rangle\langle b\rangle \neq \langle ab\rangle$?

Question

Let $R$ be an rng (no unity).

Define $IJ$ as the ideal generated by $\{ab:a\in I, b\in J\}$ for every ideals $I,J$ of $R$.

Let $I=\langle a\rangle , J=\langle b\rangle $ be principal ideals.

What would be an example that $IJ\neq \langle ab\rangle$?

Answer 1

It is even possible that $(a)(a) \neq (a^2)$, even in unital rings.

Take the ring of $2 \times 2$-matrices over any field. Consider the matrix $a = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and define $b = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$. Then one computes $a=a \cdot ba$, and this lies in $(a)(a)$. But it doesn't lie in $(a^2)$, since $a^2=0$.

In commutative rings (unital or not), we always have $(a)(b)=(ab)$.