Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero.
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 (Lam 2001, Ex.
This follows from the fact that R has a maximal left ideal M, and the fact that the quotient module R/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.
(A right full linear ring differs by using a right vector space instead.)
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}.
, 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.
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.