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January 2nd, 2015, 07:23 PM  #1 
Senior Member Joined: Aug 2014 From: India Posts: 486 Thanks: 1  An example of a compact multiplicatively unbounded ring
My teacher asked me to build an associative topological Hausdorff compact ring $\displaystyle R$ with 1, which is multiplicatively unbounded. That means there is a neighborhood $\displaystyle U∋1$ such that $\displaystyle FU≠R$ for each finite subset $\displaystyle F$ of $\displaystyle R$. 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”. 

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compact, multiplicatively, ring, unbounded 
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