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All members of the ring have an additive inverse anyway. But the set of all units in a ring constitute a group with the multiplication operation. Does anyone know why it is not required that the map defined on this group sending an element onto its inverse is not required to be continuous (when this
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The section about completion sounds a bit weird, especially the sentence "The ring S can be constructed as a set of equivalence classes of Cauchy sequences in R." It feels like this shouldn't hold in such generality (e.g. for non-separated I-adic topology).
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group inherits the subspace topology from the topology on the ring)? Or why this group is not required to be a topological group under the multiplication operation (and with the subspace topology)?
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A ring only has an inverse operation for addition, and this is the same as multiplication by -1. So the continuity of the inverse operation follows from the continuity of multiplication. --
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Does the definition of a topological ring not include the inverse operations being continuous maps (as is the case for a topological group)? --
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Some examples for the "completion" section could be given and/or developed (
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