simple ring

Ideals are the gateways through which a ring can be cut down to a quotient. A simple ring is one with no such gateways at all, apart from the two trivial ones: the zero ideal and the whole ring. There is nothing nontrivial to quotient out by, so a simple ring is, in the world of two-sided ideals, indivisible — the analogue of a prime number or a simple group.

Precisely, a simple ring is a nonzero ring R whose only two-sided ideals are 0 and R. Equivalently, every nonzero ring homomorphism out of R is injective, because the kernel would be a proper two-sided ideal. Note this controls only two-sided ideals: a simple ring can still have plenty of left ideals and right ideals, which is exactly what happens in matrix rings.

The cleanest examples are matrix rings M_n(D) over a division ring D, which are simple precisely because D has no nontrivial ideals. When one adds a finiteness hypothesis — that R is Artinian, or finite-dimensional over a field — the Artin-Wedderburn theorem proves these are the only examples: every simple Artinian ring is M_n(D) for some n and some division ring D. So simplicity plus finiteness gives a complete classification.

A caution: without finiteness the world is wilder. The Weyl algebra A_1, generated by x and d with d*x - x*d = 1 over a field of characteristic 0, is simple yet infinite-dimensional and is not a matrix ring over a division ring. Thus simple does not always mean semisimple — a simple ring need not be a sum of simple modules over itself.