A ring R is said to be a left primitive ring if it has a faithfulsimple left R-module. A right primitive ring is defined similarly with right R-modules. There are rings which are primitive on one side but not on the other. The first example was constructed by George M. Bergman in (Bergman 1964). Another example found by Jategaonkar showing the distinction can be found in Rowen (1988, p. 159).
An internal characterization of left primitive rings is as follows: a ring is left primitive if and only if there is a maximal left ideal containing no nonzero two-sided ideals. The analogous definition for right primitive rings is also valid.
One-sided primitive rings are both semiprimitive rings and prime rings. Since the product ring of two or more nonzero rings is not prime, it is clear that the product of primitive rings is never primitive.
For a left Artinian ring, it is known that the conditions "left primitive", "right primitive", "prime", and "simple" are all equivalent, and in this case it is a semisimple ring isomorphic to a square matrix ring over a division ring. More generally, in any ring with a minimal one sided ideal, "left primitive" = "right primitive" = "prime".
Every simple ringR with unity is both left and right primitive. (However, a simple non-unital ring may not be primitive.) This follows from the fact that R has a maximal left idealM, and the fact that the quotient moduleR/M is a simple left R-module, and that its annihilator is a proper two-sided ideal in R. Since R is a simple ring, this annihilator is {0} and therefore R/M is a faithful left R-module.
Weyl algebras over fields of characteristic zero are primitive, and since they are domains, they are examples without minimal one-sided ideals.
Full linear rings
A special case of primitive rings is that of full linear rings. A left full linear ring is the ring of alllinear transformations of an infinite-dimensional left vector space over a division ring. (A right full linear ring differs by using a right vector space instead.) In symbols, where V is a vector space over a division ring D. It is known that R is a left full linear ring if and only if R is von Neumann regular, left self-injective with socle soc(RR) ≠ {0}.[1] Through linear algebra arguments, it can be shown that is isomorphic to the ring of row finite matrices, where I is an index set whose size is the dimension of V over D. Likewise right full linear rings can be realized as column finite matrices over D.
Using this we can see that there are non-simple left primitive rings. By the Jacobson Density characterization, a left full linear ring R is always left primitive. When dimDV is finite R is a square matrix ring over D, but when dimDV is infinite, the set of finite rank linear transformations is a proper two-sided ideal of R, and hence R is not simple.
Goodearl, K. R. (1991), von Neumann regular rings (2 ed.), Malabar, FL: Robert E. Krieger Publishing Co. Inc., pp. xviii+412, ISBN0-89464-632-X, MR1150975
Lam, Tsi-Yuen (2001), A First Course in Noncommutative Rings, Graduate Texts in Mathematics, vol. 131 (2nd ed.), Springer, ISBN9781441986160, MR1838439
Rowen, Louis H. (1988), Ring theory. Vol. I, Pure and Applied Mathematics, vol. 127, Boston, MA: Academic Press Inc., pp. xxiv+538, ISBN0-12-599841-4, MR0940245