Let R be an integral domain Prove 1R and 1R are the only uni

Let R be an integral domain. Prove 1_R and -1_R are the only units of R that are equal to their own multiplicative inverses. Tip: Consider the equation x² - 1_R = 0_R.

Solution

If x is its own inverse then

x^2-1 =0

(x-1)(x+1)=0

But, x is in R which is a domain

HEnce,

x-1=0 or x+1=0 as there are no zero divisors in R an integral domain

So, x=1 or -1