Is a direct product $\prod_{\alpha\in{}A}S_\alpha$ of semigroups $S_\alpha$ simple, if all semigroups $S_\alpha$ are simple?

Question

I am currecntly trying to give an answer to the following problem. Consider a family of semigroups $(S_\alpha)_{\alpha\in{}A}$ and let every semigroup $S_\alpha$ be simple. Is it true or not, that then the direct product $\prod_{\alpha\in{}A}S_\alpha$ is also simple (no proper ideals)?

I haven't been able to proove this, nor haven't I been able to give a counter-example.

I would appreciate any solutions or ideas. Thanks :)

Answer 1

Yes, it is. To prove it is sufficient to consider principal ideals. Let $(a,b),(x,y),(u,v)\in S\times T$. Then $(x,y)(a,b)(u,v)=(xau,ybv)$. Since $\{xau|x,u\in S\}=S$ and $\{ybv|y,v\in T\}=T$ then the principal ideal generated by $(a,b)$ coincides with $S\times T$.