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270:(and they should have their licenses to define things revoked!) permit noncommutative rings to include commutative rings, so "strictly" is important if you are working with that definition. So I wouldn't just remove strictly, I'd ditch the whole construct and replace it with "division ring which is not a field." --
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the point is that some people might use "noncommutative" to mean "not assumed/required to be commutative". Probably not for a particular division ring, but for division rings in general. So "strictly" noncommutative would mean that, not only are we not specifying that they're commutative, but there
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Well, in context, it's about two specific division rings (the quaternions, and the rational quaternions), and it doesn't say "noncommutative ring" but rather "noncommutative division ring", which I think is clear on its own without further elaboration. I assume that anyone who uses "noncommutative
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be commutative does so because he/she takes "ring" to mean "commutative ring" by default. That's actually a reasonable choice if you use that convention (which I don't). But no one (AFAIK) uses "division ring" to mean "field" by default, so the issue doesn't come up.
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It is slightly misleading to say that French "corps" stands for
English field. First "corps" is not necessarily commutative, second, French "champ" (which literally means field) is used by French speakers to indicate a "corps commutatif", i.e., a field.
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I am French, and professional mathematician since more tham 60 years. I never heard "champ" used for "commutative field". If "champ" has had this meaning, this was before the second world war.
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I tagged the "strictly" in "strictly noncommutative" as needing clarification. I can't figure out what's meant here, nor can I even find any good sources that explain it. --
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I suspect that the intent is to say a "division ring which is not a field". The problem here seems to stem from the fact that
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If my guess is right, then I would just remove "strictly", which I don't think really clarifies anything. --
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I'm slightly guessing, or half-remembering discussions that may or may not have taken place, but I
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on
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is at least one example of elements that in fact do not commute.
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