Ring with unity - (-1)^2 = 1

Exercise 1 in section 7.1 of Dummit and Foote: Abstract Algebra is as follows:

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Let R be a ring with 1

Show that

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The web site Project Crazy Project gives the following proof:

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Let

Then

Similarly

By the uniqueness of 1,

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I am not sure why

I can see that by associativity of multiplication

But then why is (-1)x = -x ... or is this just a formal way of expressing the same thing?

Further why is (-1)(-x) = x?

Peter

Re: Ring with unity - (-1)^2 = 1

Quote:

Originally Posted by

**Bernhard** But then why is (-1)x = -x?

Thus (–1)x is the negative of x, or in other words (–1)x = –x.

Quote:

Originally Posted by

**Bernhard** Further why is (-1)(-x) = x?

From the above result with –x in place of x, it follows that (–1)(–x) = –(–x) = x.