Criterion for Ring with Unity to be Topological Ring

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\left({R, +, \circ}\right)$ be a ring with unity.

Let $\tau$ be a topology over $R$.

Suppose that $+$ and $\circ$ are $\tau$-continuous mappings.


Then $\left({R, +, \circ, \tau}\right)$ is a topological ring.


Proof

As we presume $\circ$ to be continuous, we need only prove that $\left({R, +, \tau}\right)$ is a topological group.

As we presume $+$ to be continuous, we need only show that negation is continuous.

For each $b \in R$, $- b = \left({- 1_R}\right) \circ b$ from Product with Ring Negative.

Since $\circ$ is continuous, it is continuous in each argument.

Hence negation is also continuous.

$\blacksquare$