# Difference between revisions of "Matrix ring"

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

− | The centre of $R_n$ coincides with $Z(R) E_n$, where $Z(R)$ | + | 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 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 | + | 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|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==== | ||

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</table> | </table> | ||

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## Revision as of 18:33, 12 September 2016

*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=39108