Definition:Ring of Square Matrices
Definition
Let $R$ be a ring.
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $\map {\MM_R} n$ denote the $n \times n$ matrix space over $R$.
Let $+$ denote the operation of matrix entrywise addition.
Let $\times$ be (temporarily) used to denote the operation of conventional matrix multiplication.
The algebraic structure:
- $\struct {\map {\MM_R} n, +, \times}$
is known as the ring of square matrices of order $n$ over $R$
Notation
When referring to the operation of matrix multiplication in the context of the ring of square matrices:
- $\struct {\map {\MM_R} n, +, \times}$
we must have some symbol to represent it, and $\times$ does as well as any.
However, we do not use $\mathbf A \times \mathbf B$ for matrix multiplication $\mathbf A \mathbf B$, as it is understood to mean the vector cross product, which is something completely different.
Also see
- Ring of Square Matrices over Ring is Ring
- Ring of Square Matrices over Ring with Unity
- Ring of Square Matrices over Field is Ring with Unity
- Results about rings of square matrices can be found here.