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In mathematics, a topological ring is a ring ''R'' which is also a topological space such that both the addition and the multiplication are continuous as maps :''R'' × ''R'' → ''R'', where ''R'' × ''R'' carries the product topology. == General comments == The group of units ''R''× of ''R'' is a topological group when endowed with the topology coming from the embedding of ''R''× into the product ''R'' × ''R'' as (''x'',''x''−1). However if the unit group is endowed with the subspace topology as a subspace of ''R'', it may not be a topological group, because inversion on ''R''× need not be continuous with respect to the subspace topology. An example of this situation is the adele ring of a global field; its unit group, called the idele group, is not a topological group in the subspace topology. If inversion on ''R''× is continuous in the subspace topology of ''R'' then these two topologies on ''R''× are the same. If one does not require a ring to have a unit, then one has to add the requirement of continuity of the additive inverse, or equivalently, to define the topological ring as a ring which is a topological group (for +) in which multiplication is continuous, too. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Topological ring」の詳細全文を読む スポンサード リンク
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