Abstract Algebra homework Please see there are two parts to

Abstract Algebra homework. Please see there are two parts to the question, a and b.

Prove that the multiplicative inverse of a unit a in a ring is unique. That is, if ab = ba = 1 and = = 1, then b = c. (You will need to use associativity of multiplication in R.) Indeed, more is true. If a epsilon R and there exist b, c epsilon R so a that ab = 1 and ca = 1, Prove that b = c and thus that a is a unit.

Solution

Given that ab = ba = 1 and ac = ca =1 for b not equal to c.

b = a inverse and c= a inverse

1*c = c

i.e. (ba)c = c since ba = 1

or b(ac) =c by associative property

i.e. b*1 = c or b =c

Hence if ab = ba =1 and ac = ca =1 it is necessary that b =c

Or multiplicative inverse is unique.

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b) ab = 1 and ca = 1

This gives b = a inverse and c = a inverse

Since inverse has to be unique we get b =c