An example of a compact multiplicatively unbounded ring My teacher asked me to build an associative topological Hausdorff compact ring $R$ with $1$, which is multiplicatively unbounded. That means there is a neighborhood $U\ni 1$ such that $FU\not=R$ for each finite subset $F$ of $R$. I am somewhat stuck, because I have a small stock of topological rings, and I see only two main ways to build such an example: to endow a compact topological group with a multiplication or to endow a ring with a compact ring topology. Both of these ways require a concordance of many conditions and therefore it seems to me that my success of the construction “depends on luck, but not on method”.

To make the question answered I note that a required example was given by Uri Bader here.