Let \( \displaystyle R \) be an associative ring with unity. Prove that the following are equivalent:
\( \displaystyle R \) is a local ring.
The set of non-invertible elements of \( \displaystyle R \) forms an ideal of \( \displaystyle R \).
There is a proper left ideal of \( \displaystyle R \) containing all proper left ideals of \( \displaystyle R \).
There is a proper right ideal of \( \displaystyle R \) containing all proper right ideals of \( \displaystyle R \).
Now, suppose that \( \displaystyle R \) is local and consider the set \( \displaystyle I \) of non-invertible elements of \( \displaystyle R \). Prove that
\( \displaystyle I \) is the only maximal left ideal of \( \displaystyle R \) containing all proper left ideals of \( \displaystyle R \).
\( \displaystyle I \) is the only maximal right ideal of \( \displaystyle R \) containing all proper right ideals of \( \displaystyle R \).
\( \displaystyle I \) coincides with the (Jacobson) radical \( \displaystyle J \left( R \right) \) of \( \displaystyle R \).
\( \displaystyle R/I = R/ J \left( R \right) \) is a division ring.