TITLE: $R$ has only one maximal ideal
QUESTION [5 upvotes]: Let $F$ be a field. Let $R$ be the set of all upper triangular matrices of the ring $M_{n}(F)$ with the property that its coefficients on the main diagonal are all the same. Prove that $R$ has only one maximal ideal.
Help me.

REPLY [6 votes]: The maximal ideal $I$ consists of all matrices with $0$  on the main diagonal. Indeed, it is easy to check that $I$ is an ideal. On other hand, every matrix $M\not\in I$ is invertible, so $M$ cannot contain in any proper ($\ne R$) ideal.