The center of a simple ring is necessarily a field.
It follows that a simple ring is an associative algebra over this field.
Several references (e.g., Lang (2002) or Bourbaki (2012)) require in addition that a simple ring be left or right Artinian (or equivalently semi-simple).
Under such terminology a non-zero ring with no non-trivial two-sided ideals is called quasi-simple.
), but it has nontrivial left ideals (for example, the sets of matrices which have some fixed zero columns).
matrices with entries in a division ring is simple.
Wedderburn proved these results in 1907 in his doctoral thesis, On hypercomplex numbers, which appeared in the Proceedings of the London Mathematical Society.
His thesis classified finite-dimensional simple and also semisimple algebras over fields.
One must be careful of the terminology: not every simple ring is a semisimple ring, and not every simple algebra is a semisimple algebra.
However, every finite-dimensional simple algebra is a semisimple algebra, and every simple ring that is left- or right-artinian is a semisimple ring.
An example of a simple ring that is not semisimple is the Weyl algebra.