Could somebody answer number 8 Thank you 7 if ax 1 x is a ri

Could somebody answer number (8)?

Thank you

7. if ax =1. x is a right inverse of a; if ya= 1, y is a left inverse of a. Prove that if a has a right inverse x and a left inverse y, then a is invertible, and its inverse is equal to x and to y. (First show that yaxa= 1.) 8. Prove: In a commutative ring, if ab is invertible, then a and b are both invertible.

Solution

ax =1 and ya =1

Consider yaxa = (ya) xa = 1(xa) = xa

This implies that ya is left inverse of xa

Or a inverse = yax

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8) In a commutative ring ab =ba

(ab)^-1 = b^-1a^-1

       = a^-1b^-1

a^-1   = (ab)^-1b

Hence a inverse exists.

Similarly

b^-1 = a(ab)^-1

Hence exists

That is both a and b are invertible.