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| In the branches of [[abstract algebra]] known as [[ring theory]] and [[module theory]], each right (resp. left) ''R'' module ''M'' has a '''singular submodule''' consisting of elements whose [[annihilator (ring theory)|annihilator]]s are [[essential submodule|essential]] right (resp. left) [[ideal (ring theory)|ideal]]s in ''R''. In set notation it is usually denoted as <math>\mathcal{Z}(M)=\{m\in M \mid \mathrm{ann}(m)\subseteq_e R\}\,</math>. For general rings, <math>\mathcal{Z}(M)</math> is a good generalization of the [[torsion submodule]] t(''M'') which is most often defined for [[domain (ring theory)|domain]]s. In the case that ''R'' is a commutative domain, <math>t(M)=\mathcal{Z}(M)</math>.
| | I am 34 years old and my name is Phil Greenleaf. I life in Neutraubling (Germany). |
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| If ''R'' is any ring, <math>\mathcal{Z}(R_R)</math> is defined considering ''R'' as a right module, and in this case <math>\mathcal{Z}(R_R)</math> is a twosided ideal of ''R'' called the '''right singular ideal''' of ''R''. Similarly the left handed analogue <math>\mathcal{Z}(_R R)</math> is defined. It is possible for <math>\mathcal{Z}(R_R)\neq\mathcal{Z}(_R R)</math>.
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| This article will develop several notions in terms of the singular submodule and singular ideals, including the definition of '''singular module''', '''nonsingular module''' and right and left '''nonsingular ring'''.
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| ==Definitions==
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| In the following, ''M'' is an ''R'' module:
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| *''M'' is called a '''singular module''' if <math>\mathcal{Z}(M)=M\,</math>.
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| *''M'' is called a '''nonsingular module''' if <math>\mathcal{Z}(M)=\{0\}\,</math>.
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| *''R'' is called '''right nonsingular''' if <math>\mathcal{Z}(R_R)=\{0\}\,</math>. Using the left singular ideal, a '''left nonsingular''' ring is defined similarly, and it is entirely possible for a ring to be right-not-left nonsingular.
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| In rings with unity it is always the case that <math>\mathcal{Z}(R_R)\subsetneq R\,</math>, and so "right singular ring" is not usually defined the same way as singular modules are. Some authors have used "singular ring" to mean "has a nonzero singular ideal", however this usage is not consistent with the usage of the adjectives for modules.
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| ==Properties==
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| Some general properties of the singular submodule include:
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| *<math>\mathcal{Z}(M)\cdot \mathrm{soc}(M)=\{0\}\,</math> where <math>\mathrm{soc}(M)\,</math> denotes the [[socle (mathematics)|socle]] of ''M''.
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| *If ''f'' is a homomorphism of ''R'' modules from ''M'' to ''N'', then <math>f(\mathcal{Z}(M))\subseteq \mathcal{Z}(N)\,</math>.
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| *If ''N'' is a submodule of ''M'', then <math>\mathcal{Z}(N)=N\cap \mathcal{Z}(M)\,</math>.
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| *The properties "singular" and "nonsingular" are [[Morita equivalence|Morita invariant properties]].
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| *The singular ideals of a ring contain central [[nilpotent]] elements of the ring. Consequently the singular ideal of a commutative ring contains the [[nilradical of a ring|nilradical]] of the ring.
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| *A general property of the torsion submodule is that <math>t(M/t(M))=\{0\}\,</math>, but this does not necessarily hold for the singular submodule. However if ''R'' is a right nonsingular ring, then <math>\mathcal{Z}(M/\mathcal{Z}(M))=\{0\}\,</math>.
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| *If ''N'' is an essential submodule of ''M'' (both right modules) then ''M''/''N'' is singular. If ''M'' is a [[free module]], or if ''R'' is right nonsingular, then the converse is true.
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| *A [[semisimple module]] is nonsingular if and only if it is a [[projective module]].
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| *If ''R'' is a right [[injective module#Self-injective rings|self-injective ring]], then <math>\mathcal{Z}(R_R)=J(R)\,</math>, where J(''R'') is the [[Jacobson radical]] of ''R''.
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| ==Examples==
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| Right nonsingular rings are a very broad class, including [[reduced ring]]s, and right [[Rickart ring]]s. This includes right [[hereditary ring|(semi)hereditary ring]]s, [[von Neumann regular ring]]s, [[domain (ring theory)|domain]]s, [[semisimple ring]]s, and [[Baer ring]]s.
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| For commutative rings, being nonsingular is equivalent to being a reduced ring.
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| ==Important theorems==
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| '''Johnson's Theorem''' (due to R. E. Johnson {{harv|Lam|1999|p=376}}) contains several important equivalences. For any ring ''R'', the following are equivalent:
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| # ''R'' is right nonsingular.
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| # The [[injective hull]] E(''R''<sub>''R''</sub>) is a nonsingular right ''R'' module.
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| # The endomorphism ring <math>S=\mathrm{End}(E(R_R))\,</math> is a [[semiprimitive ring]] (that is, <math>J(S)=\{0\}\,</math>).
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| # The [[maximal right ring of quotients]] <math>Q_{max}^r(R)</math> is von Neumann regular.
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| Right nonsingularity has a strong interaction with right self injective rings as well.
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| '''Theorem:''' If ''R'' is a right self injective ring, then the following conditions on ''R'' are equivalent: right nonsingular, von Neumann regular, right semihereditary, right Rickart, Baer, semiprimitive. {{harv|Lam|1999|p=262}}
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| The paper {{harv|Zelmanowitz|1983}} used nonsingular modules to characterize the class of rings whose maximal right ring of quotients have a certain structure.
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| '''Theorem:''' If ''R'' is a ring, then <math>Q_{max}^r(R)</math> is a right [[Primitive ring#Full linear rings|full linear ring]] if and only if ''R'' has a nonsingular, [[faithful module|faithful]], [[uniform module]]. Moreover, <math>Q_{max}^r(R)</math> is a finite direct product of full linear rings if and only if ''R'' has a nonsingular, faithful module with finite [[uniform dimension]].
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| == Textbooks ==
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| *{{citation |author=Goodearl, K. R. |title=Ring theory: Nonsingular rings and modules |series=Pure and Applied Mathematics, No. 33 |publisher=Marcel Dekker Inc. |place=New York |year=1976 |pages=viii+206 |mr=0429962}}
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| *{{Citation | last1=Lam | first1=Tsit-Yuen | title=Lectures on modules and rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 | mr=1653294 | year=1999}}
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| ==Primary sources==
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| *{{citation |author=Zelmanowitz, J. M. |title=The structure of rings with faithful nonsingular modules |journal=Trans. Amer. Math. Soc.
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| |volume=278 |year=1983 |number=1 |pages=347–359 |issn=0002-9947 |mr=697079 84d:16030) |doi=10.2307/1999320}}
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| <!--- Categories --->
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| [[Category:Abstract algebra]]
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| [[Category:Module theory]]
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| [[Category:Ring theory]]
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I am 34 years old and my name is Phil Greenleaf. I life in Neutraubling (Germany).