Editing Primitive ring

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In the branch of [[abstract algebra]] known as [[ring theory]], a '''left primitive ring''' is a [[Ring (mathematics)|ring]] which has a [[faithful module|faithful]] [[simple module|simple]] left [[module (mathematics)|module]]. Well known examples include [[endomorphism ring]]s of [[vector spaces]] and [[Weyl algebra]]s over [[field (mathematics)|fields]] of [[characteristic (algebra)|characteristic]] zero.

== Definition ==
A ring ''R'' is said to be a '''left primitive ring''' if it has a [[faithful module|faithful]] [[simple module|simple]] left [[module (mathematics)|''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 Bergman|George M. Bergman]] in {{harv|Bergman|1964}}.  Another example found by Jategaonkar showing the distinction can be found in {{harvtxt|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 ideal|maximal left ideal]] containing no nonzero two-sided [[ideal (ring theory)|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 [[ring homomorphism|isomorphic]] to a [[Jacobson density theorem#Topological characterization|dense]] [[subring]] of the [[ring of endomorphisms]] of a [[Division_ring#Relation_to_fields_and_linear_algebra|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 [[length of a module|finite length]] ({{harvnb|Lam|2001}}, [https://books.google.com/books?id=2T5DAAAAQBAJ&pg=PA191 Ex.&nbsp;11.19, p.&nbsp;191]).

==Properties==
One-sided primitive rings are both [[semiprimitive ring]]s and [[prime ring]]s.  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 ring|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 (mathematics)|field]].

Being left primitive is a [[Morita equivalence|Morita invariant property]].

== Examples ==
Every [[simple ring]] ''R'' with unity is both left and right primitive.  (However, a simple non-unital ring may not be primitive.)  This follows from the fact that [[maximal ideal#Properties|''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 (ring theory)|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 algebra]]s over fields of [[characteristic (algebra)|characteristic]] zero are primitive, and since they are [[domain (ring theory)|domain]]s, they are examples without minimal one-sided ideals.<!-- how about positive characteristic? -->

===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.

== See also ==
*[[primitive ideal]]

== References ==
{{reflist}}
*{{Citation | doi=10.1090/S0002-9939-1964-0167497-4 | last1=Bergman | first1=G. M. | title=A ring primitive on the right but not on the left | jstor=2034527 | mr=0167497  | year=1964 | journal=Proceedings of the American Mathematical Society | issn=0002-9939 | volume=15 | pages=473–475 | issue=3 | publisher=American Mathematical Society| doi-access=free }} [https://www.jstor.org/stable/2034929 p. 1000 errata]
*{{citation
   |last=Goodearl
   |first=K. R.
   |title=von Neumann regular rings
   |edition=2
   |publisher=Robert E. Krieger Publishing Co. Inc.
   |place=Malabar, FL
   |date=1991
   |pages=xviii+412
   |isbn=0-89464-632-X
   |mr=1150975
}}
*{{citation|title=A First Course in Noncommutative Rings|series=Graduate Texts in Mathematics|volume=131|edition=2nd|year=2001|first=Tsit-Yuen|last=Lam|publisher=Springer|isbn=9781441986160 | mr=1838439}}
*{{citation
   |last=Rowen
   |first=Louis H.
   |title=Ring theory. Vol. I
   |series=Pure and Applied Mathematics
   |volume=127
   |publisher=Academic Press Inc.
   |place=Boston, MA
   |year=1988
   |pages=xxiv+538
   |isbn=0-12-599841-4
   |mr=940245
}}

[[Category:Ring theory]]
[[Category:Algebraic structures]]