Matrix ring

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)

References

[a1]	P.M. Cohn, "Algebra" , 1–2 , Wiley (1974–1977)