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November 1

if a^2 =1 in a ring

In a Ring with identity if a^2 = 1 ,then a =1 or -1 ,i want a proof of it — Preceding unsigned comment added by 182.187.75.148 (talk) 00:14, 1 November 2012 (UTC)[reply]

What makes you so sure this is even true, if you have no proof of it? Sławomir Biały (talk) 02:01, 1 November 2012 (UTC)[reply]

sorry, there is one more condition, R has no zero divisor — Preceding unsigned comment added by 182.187.75.148 (talk) 02:20, 1 November 2012 (UTC)[reply]

If you had zero divisors, this would be false (consider, for instance,

\mathbb {Z} _{9}

; this is a ring with identity, and 5 times 2 is 1). I have a strange feeling this might just be a homework question, so I don't want to give it away, but the proof would be the same as the proof you'd see if you had a field (or even if you were just dealing with real numbers). Hope that helps. Wgunther (talk) 05:56, 1 November 2012 (UTC)[reply]

I'm not even aware that -1 makes sense for any ring with identity? In, say, $\mathbb {Z} _{n}$ you might call the number n-1 "-1" if you want to, but is this possible for any ring with identity? --KnightMove (talk) 08:45, 1 November 2012 (UTC)[reply]

Yes. -1 means the additive inverse of 1.--80.109.106.49 (talk) 10:16, 1 November 2012 (UTC)[reply]

Ok, you're right. --KnightMove (talk) 11:35, 1 November 2012 (UTC)[reply]

Well if there are no zero divisors and

(a-1)(a+1)=0

, what do you conclude about a? Sławomir Biały (talk) 23:25, 1 November 2012 (UTC)[reply]

Is there a name for the set of primes?

Is the set of primes just "the set of primes", or are there any other, preferably shorter names in use? --KnightMove (talk) 08:38, 1 November 2012 (UTC)[reply]

"Prime numbers", or possibly just "primes". — Preceding unsigned comment added by 83.70.170.48 (talk) 08:55, 1 November 2012 (UTC)[reply]

The letter "P" in blackboard bold (ℙ) is sometimes used, but this is not unambiguous. Gabbe (talk) 20:09, 1 November 2012 (UTC)[reply]

In a commutative ring, the set of prime ideals is called the "spectrum" but this is probably not what you want (even though the elements of Spec(Z) are exactly the ideals generated by prime numbers, plus the zero ideal). Rckrone (talk) 02:11, 2 November 2012 (UTC)[reply]

Indeed I mean the classic primes within the natural numbers. It seems there is no other term, thanks to all. --KnightMove (talk) 12:23, 2 November 2012 (UTC)[reply]

Einstein would have approved of this: "Things should be as simple as possible, but no simpler". -- Jack of Oz ^[Talk] 08:56, 3 November 2012 (UTC)[reply]