Matrix Ring - Properties

Properties

  • The matrix ring Mn(R) is commutative if and only if n = 1 and R is commutative. As an example for 2×2 matrices which do not commute,
\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}\,

and \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\, . This example is easily generalized to n×n matrices.

  • For n ≥ 2, the matrix ring Mn(R) has zero divisors. An example in 2×2 matrices would be
\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\, .
  • The center of a matrix ring over a ring R consists of the matrices which are scalar multiples of the identity matrix, where the scalar belongs to the center of R.
  • In linear algebra, it is noted that over a field F, Mn(F) has the property that for any two matrices A and B, AB=1 implies BA=1. 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 or sometimes weakly finite ring (Lam 1999, p. 5).

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