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I know this is a soft question of sorts but I am curious why we can't just say "1" instead of "unity," e.g. a root of unity.

J. M. ain't a mathematician
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AJJ
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    ... the concept extends to any algebraic *unital* ring. – obataku Apr 02 '13 at 21:41
  • @oldrinb: but that doesn't explain anything, since rings also have the concept of "unit", which means something quite different, making "unity" even more confusing! –  Apr 02 '13 at 21:43
  • @AsalBeagDubh unit may refer to either an element with a multiplicative inverse in some contexts, but in the context of a unital ring it refers to the multiplicative identity. – obataku Apr 02 '13 at 21:45
  • Why the downvote O_O – xavierm02 Apr 02 '13 at 21:46
  • @AsalBeagDubh in case of ambiguity, typically the multiplicative identity is referred to as *unity*. – obataku Apr 02 '13 at 21:47
  • @oldrinb: (second comment) I disagree. Wikipedia sez: http://en.wikipedia.org/wiki/Unit_%28ring_theory%29 –  Apr 02 '13 at 21:47
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    @AsalBeagDubh out of curiosity did you manage to read past the first two paragraphs? "*Unfortunately, the term unit is also used to refer to the identity element $1_R$ of the ring, in expressions like ring with a unit or unit ring, and also e.g. 'unit' matrix. (For this reason, some authors call $1_R$ "unity" or "identity", and say that $R$ is a "ring with unity" or a "ring with identity" rather than a "ring with a unit".)*" – obataku Apr 02 '13 at 21:48
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    @oldrinb: Well, I read your second comment as saying that in a unital ring, the word _unit_ **only** means _multiplicative identity_. In any case, I think this proves that the term "unity" is confusing! –  Apr 02 '13 at 21:51
  • @AsalBeagDubh yes indeed it is! :-) I figured "one" seems like a less general term than *unity*, so naming them roots of unity seems more broad a term (or a historical artifact). I apologize for my lack of clarity; I meant the term *unit* in *unital* (as in *unital ring*) refers to the multiplicative identity. – obataku Apr 02 '13 at 21:53
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    And, for those folks who *are* comfortable writing "ring with $1$" for "ring with unit(y)", why not also write "$0$-divisor" vs. "zero-divisor"? – Math Gems Apr 02 '13 at 21:55
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    @oldrinb: I get your meaning now! My two cents: I think people prefer "unity" because saying "a ring with one" sounds incomplete: the instinctive response is, "a ring with one what?" –  Apr 02 '13 at 21:56
  • @AsalBeagDubh I agree, it does sound strange... and like your previous comment pointed out, prone to confusion. – obataku Apr 02 '13 at 21:58
  • A unital ring (with invertible elements) is a ring with unity anyway so at least there is no ambiguity for the term 'unital ring', whatever you mean by 'unit'. – user50229 May 02 '13 at 21:05

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Chris Eagle
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