primitive ring

Simple rings are indivisible from the point of view of two-sided ideals, but that condition is sometimes too strong to apply directly to the rings that arise in representation theory. Primitive rings relax it just enough: instead of demanding no ideals, you demand a single faithful simple module — one irreducible representation in which no nonzero element of the ring acts as zero. A primitive ring is one that can act faithfully and irreducibly on something, the noncommutative stand-in for a field acting on a line.

Precisely, a ring R is (left) primitive if it has a faithful simple left module M; faithful means the only element of R annihilating all of M is 0, and simple means M has no submodules besides 0 and M. Equivalently, R has a maximal left ideal containing no nonzero two-sided ideal. Every simple ring is primitive, and every primitive ring is prime, so primitivity sits strictly between the two.

The structure of primitive rings is captured by the Jacobson density theorem, the noncommutative crown jewel here: a primitive ring acts on its faithful simple module M densely as a ring of linear transformations over the division ring D = End_R(M), in the sense that any finite amount of the action can be matched by an element of R. When M is finite-dimensional over D this forces R to be the full matrix ring M_n(D), recovering Artin-Wedderburn for the simple Artinian case.

Primitivity is the conceptual core of the Jacobson radical: the radical of any ring is exactly the intersection of the annihilators of all simple modules, equivalently the intersection of the kernels of all maps to primitive quotient rings. So a ring is semisimple-like precisely when it has enough primitive quotients to separate its elements, and primitive rings are the irreducible targets through which the radical is detected.