A ringR is said to be a left primitive ringif and only if it has a faithful simple 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. Another example found by Jategaonkar showing the distinction can be found in 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. The structure of left primitive rings is completely determined by the Jacobson density theorem: A ring is left primitive if and only if it is isomorphic to a dense subring of the ring of endomorphisms of a left vector space over a division ring. Another equivalent definition states that a ring is left primitive if and only if it is a prime ring with a faithful left module of finite length.
Properties
One sided primitive rings are both semiprimitive rings and prime rings. Since the ringproduct 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". A commutative ring is left primitive if and only if it is a field. Being left primitive is a Morita invariant property.
Examples
Every simple ringR with unity is both left and right primitive. This follows from the fact that R has a maximal left ideal M, and the fact that the quotient moduleR/M is a simple left R-module, and that its annihilator is a propertwo-sided ideal in R. Since R is a simple ring, this annihilator is and therefore R/M is a faithful left R-module. Weyl algebras over fields with 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. 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≠. 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.
