Matrix Entrywise Addition forms Abelian Group

Theorem
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.

Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $\struct {R, +, \circ}$.

Then $\struct {\map {\MM_R} {m, n}, +}$, where $+$ is matrix entrywise addition, is a group.

Proof
We have by definition that matrix entrywise addition is a specific instance of a Hadamard product.

By definition of a ring, the structure $\struct {R, +}$ is a group.

As $\struct {R, +}$ is a fortiori a monoid, it follows from Matrix Space Semigroup under Hadamard Product that $\struct {\map {\MM_R} {m, n}, +}$ is also a monoid.

As $\struct {R, +}$ is a group, it follows from Negative Matrix is Inverse for Matrix Entrywise Addition that all elements of $\struct {\map {\MM_R} {m, n}, +}$ have an inverse element.

From Matrix Entrywise Addition is Commutative it follows that $\struct {\map {\MM_R} {m, n}, +}$ is an Abelian group.

The result follows.

Also see

 * Hadamard Product over Group forms Group