Square matrices form a (semi)ring

Dependencies:

1. Semiring
2. Matrix
3. Matrix multiplication is associative
4. Matrix multiplication distributes over addition
5. Identity matrix

Let $R$ be a semiring. Then $\mathbb{M}_{n, n}(R)$ is a semiring. Additionally, if $R$ is a ring, then $\mathbb{M}_{n, n}(R)$ is a ring. If $R$ has a unity, then $\mathbb{M}_{n, n}(R)$ also has a unity.

Proof

• Addition is closed.
• Addition is commutative.
• Addition is associative.
• Additive identity: Zero matrix
• Multiplication is closed on square matrices.
• Multiplication is associative: See dependencies
• Distributivity: See dependencies

Therefore, $\mathbb{M}_{n, n}(R)$ is a semiring.

If $R$ is a ring, $-A$ is an additive inverse of $A$. Therefore, $\mathbb{M}_{n, n}(R)$ is a ring.

If $R$ has a unity, the identity matrix $I_n$ is a unity of $\mathbb{M}_{n, n}(R)$.

Dependency for: None

Info:

• Depth: 4
• Number of transitive dependencies: 7

Transitive dependencies:

1. Group
2. Ring
3. Semiring
4. Matrix
5. Matrix multiplication is associative
6. Matrix multiplication distributes over addition
7. Identity matrix