Editing Von Neumann regular ring

{{Short description|Rings admitting weak inverses}}
{{CS1 config|mode=cs2}}
In [[mathematics]], a '''von Neumann regular ring''' is a [[ring (mathematics)|ring]] ''R'' (associative, with 1, not necessarily commutative) such that for every element ''a'' in ''R'' there exists an ''x'' in ''R'' with {{nowrap|1=''a'' = ''axa''}}. One may think of ''x'' as a "weak inverse" of the element ''a''; in general ''x'' is not uniquely determined by ''a''.  Von Neumann regular rings are also called '''absolutely flat rings''', because these rings are characterized by the fact that every left [[module (mathematics)|''R''-module]] is [[flat module|flat]].

Von Neumann regular rings were introduced by {{harvs|txt|authorlink=John von Neumann|last=von Neumann|year=1936}} under the name of "regular rings", in the course of his study of [[von Neumann algebra]]s and [[continuous geometry]].  Von Neumann regular rings should not be confused with the unrelated [[regular ring]]s and [[regular local ring]]s of [[commutative algebra]].

An element ''a'' of a ring is called a '''von Neumann regular element''' if there exists an ''x'' such that {{nowrap|1=''a'' = ''axa''}}.{{sfn|ps=|Kaplansky|1972|p=110}}  An ideal <math>\mathfrak{i}</math> is called a (von Neumann) [[regular ideal]] if for every element ''a'' in <math>\mathfrak{i}</math> there exists an element ''x'' in <math>\mathfrak{i}</math> such that {{nowrap|1=''a'' = ''axa''}}.{{sfn|ps=|Kaplansky|1972|p=112}}

== Examples ==
Every [[field (mathematics)|field]] (and every [[skew field]]) is von Neumann regular: for {{nowrap|''a'' ≠ 0}} we can take {{nowrap|1=''x'' = ''a''<sup>−1</sup>}}.{{sfn|ps=|Kaplansky|1972|p=110}}  An [[integral domain]] is von Neumann regular if and only if it is a field.  Every [[Direct product of rings|direct product]] of von Neumann regular rings is again von Neumann regular.

Another important class of examples of von Neumann regular rings are the rings M<sub>''n''</sub>(''K'') of ''n''-by-''n'' [[square matrix|square matrices]] with entries from some field ''K''.  If ''r'' is the [[rank of a matrix|rank]] of {{nowrap|''A'' ∈ M<sub>''n''</sub>(''K'')}}, [[Gaussian elimination]] gives [[invertible matrix|invertible matrices]] ''U'' and ''V'' such that
: <math>A = U \begin{pmatrix}I_r &0\\
0 &0\end{pmatrix} V</math>
(where ''I''<sub>''r''</sub> is the ''r''-by-''r'' [[identity matrix]]). If we set {{nowrap|1=''X'' = ''V''<sup>−1</sup>''U''<sup>−1</sup>}}, then
: <math>AXA= U \begin{pmatrix}I_r &0\\
0 &0\end{pmatrix} \begin{pmatrix}I_r &0\\
0 &0\end{pmatrix} V = U \begin{pmatrix}I_r &0\\
0 &0\end{pmatrix} V = A.</math>

More generally, the {{nowrap|''n'' × ''n''}} matrix ring over any von Neumann regular ring is again von Neumann regular.{{sfn|ps=|Kaplansky|1972|p=110}}

If ''V'' is a [[vector space]] over a field (or [[Division ring|skew field]]) ''K'', then the [[endomorphism ring]] End<sub>''K''</sub>(''V'') is von Neumann regular, even if ''V'' is not finite-dimensional.{{sfn|ps=|Skornyakov|2001}}

Generalizing the above examples, suppose ''S'' is some ring and ''M'' is an ''S''-module such that every [[submodule]] of ''M'' is a direct summand of ''M'' (such modules ''M'' are called ''[[semisimple]]''). Then the [[endomorphism ring]] End<sub>''S''</sub>(''M'') is von Neumann regular. In particular, every [[semisimple ring]] is von Neumann regular. Indeed, the semisimple rings are precisely the [[Noetherian ring|Noetherian]] von Neumann regular rings.

The ring of [[affiliated operator]]s of a finite [[von Neumann algebra]] is von Neumann regular.

A [[Boolean ring]] is a ring in which every element satisfies {{nowrap|1=''a''<sup>2</sup> = ''a''}}. Every Boolean ring is von Neumann regular.

== Facts ==

The following statements are equivalent for the ring ''R'':
* ''R'' is von Neumann regular
* every [[principal ideal|principal]] [[left ideal]] is generated by an [[idempotent element (ring theory)|idempotent element]]
* every [[finitely generated module|finitely generated]] left ideal is generated by an idempotent
* every principal left ideal is a [[direct summand]] of the left ''R''-module ''R''
* every finitely generated left ideal is a direct summand of the left ''R''-module ''R''
* every finitely generated [[submodule]] of a [[projective module|projective]] left ''R''-module ''P'' is a direct summand of ''P''
* every left ''R''-module is [[flat module|flat]]: this is also known as ''R'' being '''absolutely flat''', or ''R'' having '''[[weak dimension]]''' 0
* every [[short exact sequence]] of left ''R''-modules is [[pure exact]].
The corresponding statements for right modules are also equivalent to ''R'' being von Neumann regular.

Every von Neumann regular ring has [[Jacobson radical]] {0} and is thus [[semiprimitive ring|semiprimitive]] (also called "Jacobson semi-simple").

In a commutative von Neumann regular ring, for each element ''x'' there is a unique element ''y'' such that {{nowrap|1=''xyx'' = ''x''}} and {{nowrap|1=''yxy'' = ''y''}}, so there is a canonical way to choose the "weak inverse" of ''x''.

The following statements are equivalent for the commutative ring ''R'':
* ''R'' is von Neumann regular.
* ''R'' has [[Krull dimension]] 0 and is [[reduced ring|reduced]].
* Every [[localization of a ring|localization]] of ''R'' at a [[maximal ideal]] is a field.
* ''R'' is a subring of a product of fields closed under taking "weak inverses" of {{nowrap|''x'' ∈ ''R''}} (the unique element ''y'' such that {{nowrap|1=''xyx'' = ''x''}} and {{nowrap|1=''yxy'' = ''y''}}).
* ''R'' is a [[V-ring (ring theory)|V-ring]].{{sfn|ps=|Michler|Villamayor|1973}}
* ''R'' has the [[Right-lifting-property|right-lifting property]] against the ring homomorphism {{nowrap|'''Z'''[''t''] → '''Z'''[''t''<sup>±</sup>] × '''Z'''}} determined by {{nowrap|''t'' ↦ (''t'', 0)}}, or said geometrically, every [[regular function]] <math display="inline">\mathrm{Spec}(R) \to \mathbb{A}^1</math> factors through the [[morphism of schemes]] <math>\{0\} \sqcup \mathbb{G}_m \to \mathbb{A}^1</math>.{{sfn|ps=|Burklund|Schlank|Yuan|2022}}

Also, the following are equivalent: for a commutative ring ''A''
* {{nowrap|1=''R'' = ''A'' / nil(''A'')}} is von Neumann regular.
* The [[spectrum of a ring|spectrum]] of ''A'' is Hausdorff (in the [[Zariski topology]]).
* The [[constructible topology]] and Zariski topology for [[Spectrum of a ring|Spec(''A'')]] coincide.

== Generalizations and specializations ==
{{Unreferenced section|date=August 2023}}

Special types of von Neumann regular rings include ''unit regular rings'' and ''strongly von Neumann regular rings'' and [[rank ring]]s.

A ring ''R'' is called '''unit regular''' if for every ''a'' in ''R'', there is a unit ''u'' in ''R'' such that {{nowrap|1=''a'' = ''aua''}}.  Every [[semisimple ring]] is unit regular, and unit regular rings are [[directly finite ring]]s. An ordinary von Neumann regular ring need not be directly finite.

A ring ''R'' is called '''strongly von Neumann regular''' if for every ''a'' in ''R'', there is some ''x'' in ''R'' with {{nowrap|1=''a'' = ''aax''}}.  The condition is left-right symmetric.  Strongly von Neumann regular rings are unit regular.  Every strongly von Neumann regular ring is a [[subdirect product]] of [[division ring]]s.  In some sense, this more closely mimics the properties of commutative von Neumann regular rings, which are subdirect products of fields.  For commutative rings, von Neumann regular and strongly von Neumann regular are equivalent.  In general, the following are equivalent for a ring ''R'':
* ''R'' is strongly von Neumann regular
* ''R'' is von Neumann regular and [[reduced ring|reduced]]
* ''R'' is von Neumann regular and every idempotent in ''R'' is [[idempotent (ring theory)#Types of ring idempotents|central]]
* Every principal left ideal of ''R'' is generated by a central idempotent

Generalizations of von Neumann regular rings include '''π'''-regular rings, left/right [[semihereditary ring]]s, left/right [[nonsingular ring]]s and [[semiprimitive ring]]s.

== See also ==
* [[Regular semigroup]]
* [[Weak inverse]]

== Notes ==
{{reflist}}

== References ==
{{refbegin}}
* {{cite arXiv
  | last1=Burklund |first1=Robert
  | last2=Schlank |first2=Tomer M.
  | last3=Yuan |first3=Allen
  | date=2022-07-20
  | title=The Chromatic Nullstellensatz
  | eprint=2207.09929
  | page=50
  | class=math.AT
  }}
* {{citation
  | last=Kaplansky | first=Irving | authorlink=Irving Kaplansky
  | year=1972
  | title=Fields and rings
  | edition=Second | series=Chicago lectures in mathematics
  | publisher=University of Chicago Press
  | isbn=0-226-42451-0 | zbl=1001.16500
  }}
* {{cite journal
  | last1=Michler |first1=G.O.
  | last2=Villamayor |first2=O.E.
  | date=April 1973
  | title=On rings whose simple modules are injective
  | journal=[[Journal of Algebra]]
  | volume=25 | issue=1 | pages=185–201
  | doi=10.1016/0021-8693(73)90088-4 | doi-access=free
  | hdl=20.500.12110/paper_00218693_v25_n1_p185_Michler | hdl-access=free
  }}
* {{sfn whitelist|CITEREFSkornyakov2001}}{{eom
  | id=R/r080830
  | author-last1=Skornyakov |author-first1=L.A.
  | title=Regular ring (in the sense of von Neumann)
   }}
* {{citation
  | last1=von Neumann | first1=John | author1-link=John von Neumann
  | year=1936
  | title=On Regular Rings
  | doi=10.1073/pnas.22.12.707 | jfm=62.1103.03
  | journal=Proc. Natl. Acad. Sci. USA
  | volume=22 | issue=12 | pages=707–713
  | pmid=16577757 | pmc=1076849 | bibcode=1936PNAS...22..707V | zbl=0015.38802  | doi-access=free
  }}
{{refend}}

== Further reading ==
{{refbegin}}
* {{citation
  |last1=Goodearl |first1=K. R.
  |year=1991
  |title=von Neumann regular rings
  |edition=2
  |place=Malabar, FL  |publisher=Robert E. Krieger Publishing Co. Inc.
  |pages=xviii+412
  |isbn=0-89464-632-X  |mr=1150975 | zbl=0749.16001
  }}
* {{citation
  | last1=von Neumann | first1=John | author1-link=John von Neumann
  | year=1960
  | title=Continuous geometries
  | publisher=[[Princeton University Press]]
  | zbl=0171.28003
  }}
{{refend}}

[[Category:Ring theory]]
[[Category:John von Neumann]]