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| In [[mathematics]], a '''von Neumann regular ring''' is a [[ring (mathematics)|ring]] ''R'' such that for every ''a'' in ''R'' there exists an ''x'' in ''R'' such that ''a'' = ''axa''. To avoid the possible confusion with the [[regular ring]]s and [[regular local ring]]s of [[commutative algebra]] (which are unrelated notions), von Neumann regular rings are also called '''absolutely flat rings''', because these rings are characterized by the fact that every left [[module (mathematics)|module]] is [[flat module|flat]].
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| One may think of ''x'' as a "weak inverse" of ''a''. In general ''x'' is not uniquely determined by ''a''.
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| Von Neumann regular rings were introduced by {{harvs|txt|authorlink=John von Neumann|last=von Neumann|year=1936}} under the name of "regular rings", during his study of [[von Neumann algebra]]s and [[continuous geometry]].
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| An element ''a'' of a ring is called a '''von Neumann regular element''' if there exists an ''x'' such that ''a''=''axa''.<ref name=K110>Kaplansky (1972) p.110</ref> An ideal <math>\mathfrak{i}</math> is called a (von Neumann) [[regular ideal]] if it is a von Neumann regular non-unital ring, i.e if for every element ''a'' in <math>\mathfrak{i}</math> there exists an element ''x'' in <math>\mathfrak{i}</math> such that ''a''=''axa''.<ref name=K112>Kaplansky (1972) p.112</ref>
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| == Examples ==
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| Every [[field (mathematics)|field]] (and every [[skew field]]) is von Neumann regular: for ''a''≠0 we can take ''x'' = ''a''<sup> -1</sup>.<ref name=K110/> An [[integral domain]] is von Neumann regular if and only if it is a field.
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| Another example of a von Neumann regular ring is the ring 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 ''A''∈M<sub>''n''</sub>(''K''), then there exist [[invertible matrix|invertible matrices]] ''U'' and ''V'' such that
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| :<math>A = U \begin{pmatrix}I_r &0\\ | |
| 0 &0\end{pmatrix} V</math>
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| (where ''I''<sub>''r''</sub> is the ''r''-by-''r'' [[identity matrix]]). If we set ''X'' = ''V''<sup> -1</sup>''U''<sup> -1</sup>, then
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| :<math>AXA= U \begin{pmatrix}I_r &0\\
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| 0 &0\end{pmatrix} \begin{pmatrix}I_r &0\\
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| 0 &0\end{pmatrix} V = U \begin{pmatrix}I_r &0\\
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| 0 &0\end{pmatrix} V = A.</math>
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| More generally, the matrix ring over a von Neumann regular ring is again a von Neumann regular ring.<ref name=K110/>
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| The ring of [[affiliated operator]]s of a finite [[von Neumann algebra]] is von Neumann regular.
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| A [[Boolean ring]] is a ring in which every element satisfies ''a''<sup>2</sup> = ''a''. Every Boolean ring is von Neumann regular.
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| == Facts == | |
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| The following statements are equivalent for the ring ''R'':
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| * ''R'' is von Neumann regular
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| * every [[principal ideal|principal]] [[left ideal]] is generated by an [[idempotent element]]
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| * every [[finitely generated module|finitely generated]] left ideal is generated by an idempotent
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| * every principal left ideal is a [[direct summand]] of the left ''R''-module ''R''
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| * every finitely generated left ideal is a direct summand of the left ''R''-module ''R''
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| * every finitely generated [[submodule]] of a [[projective module|projective]] left ''R''-module ''P'' is a direct summand of ''P''
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| * every left ''R''-module is [[flat module|flat]]: this is also known as ''R'' being '''absolutely flat''', or ''R'' having '''weak dimension''' 0.
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| * every [[short exact sequence]] of left ''R''-modules is [[pure exact]]
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| The corresponding statements for right modules are also equivalent to ''R'' being von Neumann regular.
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| In a commutative von Neumann regular ring,
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| for each element ''x'' there is a unique element ''y'' such that ''xyx''=''x'' and ''yxy''=''y'', so there is a canonical way to choose the "weak inverse" of ''x''.
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| The following statements are equivalent for the commutative ring ''R'':
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| * ''R'' is von Neumann regular
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| * ''R'' has [[Krull dimension]] 0 and is [[reduced ring|reduced]]
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| * Every [[localization of a ring|localization]] of ''R'' at a [[maximal ideal]] is a field
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| *''R'' is a subring of a product of fields closed under taking "weak inverses" of ''x''∈''R'' (the unique element ''y'' such that ''xyx''=''x'' and ''yxy''=''y'').
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| Also, the following are equivalent: for a commutative ring ''A''
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| * <math>R=A / nil(A)</math> is von Neumann regular.
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| * The [[spectrum of a ring|spectrum]] of ''R'' is Hausdorff (with respect to Zariski topology).
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| * The [[constructible topology]] and Zariski topology for <math>Spec(A)</math> coincide.
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| Every [[semisimple ring]] is von Neumann regular, and a left (or right) [[Noetherian ring|Noetherian]] von Neumann regular ring is semisimple. Every von Neumann regular ring has [[Jacobson radical]] {0} and is thus [[semiprimitive ring|semiprimitive]] (also called "Jacobson semi-simple").
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| Generalizing the above example, 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.
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| == Generalizations and specializations==
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| Special types of von Neumann regular rings include ''unit regular rings'' and ''strongly von Neumann regular rings'' and [[rank ring]]s.
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| A ring ''R'' is called '''unit regular''' if for every ''a'' in ''R'', there is a unit ''u'' in ''R'' such that ''a=aua''. Every [[semisimple ring]] ring is unit regular, and unit regular rings are [[directly finite ring]]s. An ordinary von Neumann regular ring need not be directly finite.
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| A ring ''R'' is called '''strongly von Neumann regular''' if for every ''a'' in ''R'', there is some ''x'' in ''R'' with ''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. Of course for commutative rings, von Neumann regular and strongly von Neumann regular are equivalent. In general, the following are equivalent for a ring ''R'':
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| * ''R'' is strongly von Neumann regular
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| * ''R'' is von Neumann regular and [[reduced ring|reduced]]
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| * ''R'' is von Neumann regular and every idempotent in ''R'' is central
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| * Every principal left ideal of ''R'' is generated by a central idempotent
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| Generalizations of von Neumann regular rings include '''π'''-regular rings, left/right [[semihereditary ring]]s, left/right [[nonsingular ring]]s and [[semiprimitive ring]]s.
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| == See also ==
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| * [[Regular semigroup]]
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| * [[Weak inverse]]
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| ==References==
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| {{reflist}}
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| * {{citation | first=Irving | last=Kaplansky | authorlink=Irving Kaplansky | title=Fields and rings | edition=Second | series=Chicago lectures in mathematics | publisher=University of Chicago Press | year=1972 | isbn=0-226-42451-0 | zbl=1001.16500 }}
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| == Further reading ==
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| *{{citation |author=Goodearl, K. R. |title=von Neumann regular rings |edition=2 |publisher=Robert E. Krieger Publishing Co. Inc. |place=Malabar, FL |year=1991 |pages=xviii+412 |isbn=0-89464-632-X |mr=1150975 | zbl=0749.16001 }}
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| *{{eom|id=R/r080830|title=Regular ring (in the sense of von Neumann)|author=L.A. Skornyakov}}
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| *{{Citation | last1=von Neumann | first1=John | author1-link=John von Neumann | title=On Regular Rings | doi=10.1073/pnas.22.12.707 | jfm=62.1103.03 | year=1936 | journal=Proc. Nat. Acad. Sci. USA | volume=22 | pages=707–712 | pmid=16577757 | issue=12 | pmc=1076849 | zbl=0015.38802 }}
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| *{{Citation | last1=von Neumann | first1=John | author1-link=John von Neumann | title=Continuous geometries | publisher=[[Princeton University Press]] | year=1960 | zbl=0171.28003 }}
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| [[Category:Ring theory]]
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