Prove that if every element in a ring R except 1 has a left quasi-inverse, then R is

Hello !

can you help me proving this exercise please :

An element a in a ring R has a left quasi inverse if there exist an element b in R with a+b-ab=0.

Prove that if every *element* in a ring R except 1 has a left *quasi*-*inverse*, then R is

a *division ring*..

THANK YOU !

Re: Prove that if every element in a ring R except 1 has a left quasi-inverse, then R

In other words, show that if every element, other than 1, has a left quasi-inverse, then every element, other than 0, has an inverse.

I take it we are allowed to **assume** that the ring **has** a multiplicative identity, 1?

Re: Prove that if every element in a ring R except 1 has a left quasi-inverse, then R

here is something to get you started:

a + b - ab = 0

a + (1 - a)b = 0

1 + (1 - a)b = 1 - a

1 = (1 - a) - (1 - a)b

1 = (1 - a)(1 - b).

so 1 - a has a right-inverse, unless a = 1. why does this show U(R) = R\{0}?

Re: Prove that if every element in a ring R except 1 has a left quasi-inverse, then R

Thank you very much for your help