Note: A simple Ring is a Ring such that it hasn't got proper ideals.
I can prove that all division Ring is simple, but, is true that all simple ring is a division ring? If not, can you give a counter example?
Thanks!
an easy example is the ring of all matrices with entries in a field . it's obviously not a division ring because not every non-zero matrix is invertible.
the reason that is simple is this well-known and easy to prove fact that for any ring the two-sided ideals of are exactly in the form where is a two-sided
ideal of thus is simple if and only if is simple.