Let $R$ be a commutative ring with identity 1, such that $R$ contains exactly three ideals:$\{0\},J,R$. Prove that $R$ is not an integral domain.  I was thinking start with supposing that R is an integral domain, then for $ab=0, a=0,b \ne0$ for $b \in R-J$, it can found a contradiction of proving $Rb$ is also  an ideal. But I cannot go on anymore, may be I was thinking wrong and hope some body can help me with it. Thanks
Let $R$ be a commutative ring with identity. Assume that $R$ has exactly three distinct ideals: $\{0\},I, R.$      1) Show that if $a \in R-I$, then $a$ is a unit in $R$.      2) Let $a,b\ne0$ in $I$. Show that $ab=0$.
proving combinatorics identity - $\sum_{k=0}^m{n-k \choose m-k}={n+1 \choose m}$ I have just bought a new bike and had it about a month now. I don't really know much about bikes but I do know how to repair an inner tube.  I have had one puncture and just bought a new inner tube. It worked for a few days and then went down again so this time I got a puncture repair kit. I took the inner tube out to check for any punctures and found nothing. I put it in water and pumped it up and there were no bubbles. I went around the entire tube pushing it next to my ear and heard nothing. I checked the tire to see if anything was wrong - there was nothing that I could see.   So I put it all back together and took it for a ride and got about half a mile and the tire was flat. I pumped it up and got about another half a mile and it was flat again.  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