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

Could somebody answer number (7)?

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

given that a has a right inverse x.

This means ax = 1.

Similarly a has a left inverse y implies that

ya =1

ax =1 or a = 1(x^-1)

Also a = y^-1(1)

a(xy) = ax(y) = 1(y) = y = y(a)x

But yaxa =1 this means yax is inverse of a.

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8) we have that ab inverse = b^-1a^-1

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

As ab is commutative we have ab = ba

Hence inverse of ba = b^-1a^-1

Or multiply both sides by b.

bba = a^-1 Thus it follows that a^-1 exists and equal to bba.

Similarly b^-1 = baa,

So a and b are invertible.