# Math Help - Simple Ring

1. ## Simple Ring

Note: A simple Ring is a Ring $A \neq {0}$ 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!

2. Originally Posted by roporte
Note: A simple Ring is a Ring $A \neq {0}$ 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 $A=M_n(F), \ n \geq 2,$ the ring of all $n \times n$ matrices with entries in a field $F$. it's obviously not a division ring because not every non-zero matrix is invertible.

the reason that $A$ is simple is this well-known and easy to prove fact that for any ring $R$ the two-sided ideals of $M_n(R)$ are exactly in the form $M_n(I),$ where $I$ is a two-sided

ideal of $R.$ thus $M_n(R)$ is simple if and only if $R$ is simple.