Editing Matrix ring (section)

== Properties ==
* If ''S'' is a [[subring]] of ''R'', then M<sub>''n''</sub>(''S'') is a subring of M<sub>''n''</sub>(''R''). For example, M<sub>''n''</sub>('''Z''') is a subring of M<sub>''n''</sub>('''Q''').
* The matrix ring M<sub>''n''</sub>(''R'') is [[commutative ring|commutative]] if and only if {{nowrap|1=''n'' = 0}}, {{nowrap|1=''R'' = 0}}, or ''R'' is [[commutative ring|commutative]] and {{nowrap|1=''n'' = 1}}. In fact, this is true also for the subring of upper triangular matrices.  Here is an example showing two upper triangular {{nowrap|2 × 2}} matrices that do not commute, assuming {{nowrap| 1 ≠ 0}} in ''R'':
*:: <math>
  \begin{bmatrix}
    1 & 0 \\
    0 & 0 
  \end{bmatrix}
  \begin{bmatrix}
    1 & 1 \\
    0 & 0
  \end{bmatrix}
=
  \begin{bmatrix}
    1 & 1 \\
    0 & 0
  \end{bmatrix}
</math>
*: and
*:: <math>  \begin{bmatrix}
    1 & 1 \\
    0 & 0 
  \end{bmatrix}
  \begin{bmatrix}
    1 & 0 \\
    0 & 0
  \end{bmatrix}
=
  \begin{bmatrix}
    1 & 0 \\
    0 & 0
  \end{bmatrix}.
</math>
* For {{nowrap|''n'' ≥ 2}}, the matrix ring M<sub>''n''</sub>(''R'') over a [[zero ring|nonzero ring]] has [[zero divisor]]s and [[nilpotent element]]s; the same holds for the ring of upper triangular matrices. An example in {{nowrap|2 × 2}} matrices would be
*:: <math>  \begin{bmatrix}
    0 & 1 \\
    0 & 0 
  \end{bmatrix}
  \begin{bmatrix}
    0 & 1 \\
    0 & 0
  \end{bmatrix}
=
  \begin{bmatrix}
    0 & 0 \\
    0 & 0
  \end{bmatrix}.
</math>
* The [[center (ring theory)|center]] of M<sub>''n''</sub>(''R'') consists of the scalar multiples of the [[identity matrix]], ''I''<sub>''n''</sub>, in which the scalar belongs to the center of ''R''.
* The [[unit group]] of M<sub>''n''</sub>(''R''), consisting of the invertible matrices under multiplication, is denoted GL<sub>''n''</sub>(''R'').
* If ''F'' is a field, then for any two matrices ''A'' and ''B'' in M<sub>''n''</sub>(''F''), the equality {{nowrap|1=''AB'' = ''I''<sub>''n''</sub>}} implies {{nowrap|1=''BA'' = ''I''<sub>''n''</sub>}}.  This is not true for every ring ''R'' though.  A ring ''R'' whose matrix rings all have the mentioned property is known as a [[stably finite ring]] {{harv|Lam|1999|p=5}}.