Editing Primitive ring (section)

===Full linear rings===
A special case of primitive rings is that of ''full linear rings''.  A '''left full linear ring''' is the ring of ''all'' [[linear transformation]]s 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, <math>R=\mathrm{End}(_D V)</math> 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]], [[Injective module#Self-injective rings|left self-injective]] with [[socle (mathematics)|socle]] soc(<sub>''R''</sub>''R'') ≠ {0}.{{sfn|Goodearl|1991|p=100}} Through [[linear algebra]] arguments, it can be shown that <math>\mathrm{End}(_D V)\,</math> is isomorphic to the ring of [[matrix ring#Examples|row finite matrices]] <math>\mathbb{RFM}_I(D)\,</math>, 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 dim<sub>''D''</sub>''V'' is finite ''R'' is a square matrix ring over ''D'', but when dim<sub>''D''</sub>''V'' is infinite, the set of finite rank linear transformations is a proper two-sided ideal of ''R'', and hence ''R'' is not simple.