topological ring

A ring $R$ which is a topological space is called a topological ring if the addition, multiplication, and the additive inverse functions are continuous functions from $R\\times R$ to $R$ .

A topological division ring is a topological ring such that the multiplicative inverse function is continuous away from $0$ . A topological field is a topological division ring that is a field.

Remark. It is easy to see that if $R$ contains the multiplicative identity $1$ , then $R$ is a topological ring iff addition and multiplication are continuous. This is true because the additive inverse of an element can be written as the product of the element and $-1$ . However, if $R$ does not contain $1$ , it is necessary to impose the continuity condition on the additive inverse operation.

单词	TopologicalRing
释义	topological ring A ring $R$ which is a topological space is called a topological ring if the addition, multiplication, and the additive inverse functions are continuous functions from $R\\times R$ to $R$ . A topological division ring is a topological ring such that the multiplicative inverse function is continuous away from $0$ . A topological field is a topological division ring that is a field. Remark. It is easy to see that if $R$ contains the multiplicative identity $1$ , then $R$ is a topological ring iff addition and multiplication are continuous. This is true because the additive inverse of an element can be written as the product of the element and $-1$ . However, if $R$ does not contain $1$ , it is necessary to impose the continuity condition on the additive inverse operation.
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