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''full matrix ring''
 
''full matrix ring''
  
The ring of all square matrices of a fixed order over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m0628501.png" />. The ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m0628502.png" />-dimensional matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m0628503.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m0628504.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m0628505.png" />. Throughout this article <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m0628506.png" /> is an associative ring with identity (cf. [[Associative rings and algebras|Associative rings and algebras]]).
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The ring of all square matrices of a fixed order over a ring $R$, with the operations of [[matrix addition]] and [[matrix multiplication]]. The ring of $(n \times n)$-dimensional matrices over $R$ is denoted by $R_n$ or $M_n(R)$. Throughout this article $R$ is an [[Associative rings and algebras|associative ring]] [[unital ring|with identity]].
  
The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m0628507.png" /> is isomorphic to the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m0628508.png" /> of all endomorphisms of the free right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m0628509.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285010.png" />, possessing a basis with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285011.png" /> elements. The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285012.png" /> is the identity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285013.png" />. An associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285014.png" /> with identity 1 is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285015.png" /> if and only if there is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285016.png" /> a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285017.png" /> elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285019.png" />, subject to the following conditions:
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The ring $R_n$ is isomorphic to the ring $\mathop{End}(M)$ of all endomorphisms of the free right $R$-module $M = R^n$, possessing a basis with $n$ elements. The [[identity matrix]] $E_n = \text{diag}(1,\ldots,1)$ is the identity in $R_n$. An associative ring $A$ with identity 1 is isomorphic to $R_n$ if and only if there is in $A$ a set of $n^2$ elements $e_{ij}$, $i,j=1,\ldots,n$, subject to the following conditions:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285021.png" />;
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1) $e_{ij}e_{kl} = \delta_{jk} e_{il}$, $\sum_{i=1}^n e_{ii}e_{ii} = 1$;
  
2) the centralizer of the set of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285022.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285023.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285024.png" />.
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2) the centralizer of the set of elements $e_{ij}$ in $A$ is isomorphic to $R$.
  
The centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285025.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285027.png" /> is the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285028.png" />; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285029.png" /> the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285030.png" /> is non-commutative.
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The centre of $R_n$ coincides with $\mathcal{Z}(R) E_n$, where $\mathcal{Z}(R)$ denotes the centre of $R$; for $n>1$ the ring $R_n$ is non-commutative.
  
The multiplicative group of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285031.png" /> (the group of all invertible elements), called the [[General linear group|general linear group]], is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285032.png" />. A matrix from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285033.png" /> is invertible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285034.png" /> if and only if its columns form a basis of the free right module of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285035.png" />-dimensional matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285036.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285037.png" /> is commutative, then the invertibility of a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285038.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285039.png" /> is equivalent to the invertibility of its determinant, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285040.png" />, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285041.png" />. The equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285042.png" /> holds.
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The multiplicative group of the ring $R_n$ (the group of all invertible elements), called the [[general linear group]], is denoted by $\mathop{GL}_n(R)$. A matrix from $R_n$ is invertible in $R_n$ if and only if its columns form a basis of the free right module of all $(n \times 1)$-dimensional matrices over $R$. If $R$ is commutative, then the [[determinant]] is defined as a multiplicative map from $R_n$ to $R$ and invertibility of a matrix $X$ in $R_n$ is equivalent to the invertibility of its determinant, $\det X$, in $R$. The isomorphism $R_{mn} \sim (R_m)_n$ holds.
  
The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285043.png" /> is simple if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285044.png" /> is simple, for the two-sided ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285045.png" /> are of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285046.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285047.png" /> is a two-sided ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285048.png" />. An [[Artinian ring|Artinian ring]] is simple if and only if it is isomorphic to a matrix ring over a skew-field (the Wedderburn–Artin theorem). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285049.png" /> denotes the [[Jacobson radical|Jacobson radical]] of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285050.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285051.png" />. Consequently, every matrix ring over a semi-simple ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285052.png" /> is semi-simple. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285053.png" /> is regular (i.e. if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285054.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285055.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285056.png" />), then so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285057.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285058.png" /> is a ring with an invariant basis number, i.e. the number of elements in a basis of each free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285059.png" />-module does not depend of the choice of the basis, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285060.png" /> also has this property. The rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285062.png" /> are equivalent in the sense of Morita (see [[Morita equivalence|Morita equivalence]]): The category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285063.png" />-modules is equivalent to the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285064.png" />-modules. However, the fact that projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285065.png" />-modules are free does not necessarily entail that projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285066.png" />-modules are free too. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285067.png" /> is a field and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285068.png" />, then there exist finitely-generated projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062850/m06285069.png" />-modules which are not free.
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The two-sided ideals in $R_n$ are of the form $J_n$, where $J$ is a two-sided ideal in $R$ and so the ring $R_n$ is [[simple ring|simple]] if and only if $R$ is simple. An [[Artinian ring]] is simple if and only if it is isomorphic to a matrix ring over a [[skew-field]] (the [[Wedderburn–Artin theorem]]). If $\mathcal{J}(R)$ denotes the [[Jacobson radical]] of the ring $R$, then $\mathcal{J}(R_n) = \mathcal{J}(R)_n$. Consequently, every matrix ring over a [[semi-simple ring]] $R$ is semi-simple. If $R$ is a [[regular ring (in the sense of von Neumann)]] (i.e. if for every $a \in R$ there is a $b \in R$ such that $aba = a$), then so is $R_n$. If $R$ is a ring with an invariant basis number, i.e. the number of elements in a basis of each free $R$-module does not depend on the choice of the basis, then $R_n$ also has this property. The rings $R$ and $R_n$ are equivalent in the sense of Morita (see [[Morita equivalence]]): The category of $R$-modules is equivalent to the category of $R_n$-modules. However, the condition that projective $R$-modules are free does not necessarily entail that projective $R_n$-modules are free too. For instance, if $R$ is a field and $n>1$, then there exist finitely-generated projective $R_n$-modules which are not free.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Faith,  "Algebra: rings, modules, and categories" , '''1''' , Springer  (1973)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Lambek,  "Lectures on rings and modules" , Blaisdell  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.A. Bokut',  "Associative rings" , '''1''' , Novosibirsk  (1977)  (In Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  C. Faith,  "Algebra: rings, modules, and categories" , '''1''' , Springer  (1973)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  J. Lambek,  "Lectures on rings and modules" , Blaisdell  (1966)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  L.A. Bokut',  "Associative rings" , '''1''' , Novosibirsk  (1977)  (In Russian)</TD></TR>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,  "Algebra" , '''1–2''' , Wiley  (1974–1977)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,  "Algebra" , '''1–2''' , Wiley  (1974–1977)</TD></TR>
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</table>
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{{TEX|done}}

Latest revision as of 06:20, 13 September 2016

2010 Mathematics Subject Classification: Primary: 16S50 [MSN][ZBL]

full matrix ring

The ring of all square matrices of a fixed order over a ring $R$, with the operations of matrix addition and matrix multiplication. The ring of $(n \times n)$-dimensional matrices over $R$ is denoted by $R_n$ or $M_n(R)$. Throughout this article $R$ is an associative ring with identity.

The ring $R_n$ is isomorphic to the ring $\mathop{End}(M)$ of all endomorphisms of the free right $R$-module $M = R^n$, possessing a basis with $n$ elements. The identity matrix $E_n = \text{diag}(1,\ldots,1)$ is the identity in $R_n$. An associative ring $A$ with identity 1 is isomorphic to $R_n$ if and only if there is in $A$ a set of $n^2$ elements $e_{ij}$, $i,j=1,\ldots,n$, subject to the following conditions:

1) $e_{ij}e_{kl} = \delta_{jk} e_{il}$, $\sum_{i=1}^n e_{ii}e_{ii} = 1$;

2) the centralizer of the set of elements $e_{ij}$ in $A$ is isomorphic to $R$.

The centre of $R_n$ coincides with $\mathcal{Z}(R) E_n$, where $\mathcal{Z}(R)$ denotes the centre of $R$; for $n>1$ the ring $R_n$ is non-commutative.

The multiplicative group of the ring $R_n$ (the group of all invertible elements), called the general linear group, is denoted by $\mathop{GL}_n(R)$. A matrix from $R_n$ is invertible in $R_n$ if and only if its columns form a basis of the free right module of all $(n \times 1)$-dimensional matrices over $R$. If $R$ is commutative, then the determinant is defined as a multiplicative map from $R_n$ to $R$ and invertibility of a matrix $X$ in $R_n$ is equivalent to the invertibility of its determinant, $\det X$, in $R$. The isomorphism $R_{mn} \sim (R_m)_n$ holds.

The two-sided ideals in $R_n$ are of the form $J_n$, where $J$ is a two-sided ideal in $R$ and so the ring $R_n$ is simple if and only if $R$ is simple. An Artinian ring is simple if and only if it is isomorphic to a matrix ring over a skew-field (the Wedderburn–Artin theorem). If $\mathcal{J}(R)$ denotes the Jacobson radical of the ring $R$, then $\mathcal{J}(R_n) = \mathcal{J}(R)_n$. Consequently, every matrix ring over a semi-simple ring $R$ is semi-simple. If $R$ is a regular ring (in the sense of von Neumann) (i.e. if for every $a \in R$ there is a $b \in R$ such that $aba = a$), then so is $R_n$. If $R$ is a ring with an invariant basis number, i.e. the number of elements in a basis of each free $R$-module does not depend on the choice of the basis, then $R_n$ also has this property. The rings $R$ and $R_n$ are equivalent in the sense of Morita (see Morita equivalence): The category of $R$-modules is equivalent to the category of $R_n$-modules. However, the condition that projective $R$-modules are free does not necessarily entail that projective $R_n$-modules are free too. For instance, if $R$ is a field and $n>1$, then there exist finitely-generated projective $R_n$-modules which are not free.

References

[1] C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973)
[2] J. Lambek, "Lectures on rings and modules" , Blaisdell (1966)
[3] L.A. Bokut', "Associative rings" , 1 , Novosibirsk (1977) (In Russian)


Comments

References

[a1] P.M. Cohn, "Algebra" , 1–2 , Wiley (1974–1977)
How to Cite This Entry:
Matrix ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matrix_ring&oldid=15509
This article was adapted from an original article by D.A. Suprunenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article