# Answer to Question #17266 in Abstract Algebra for Tsit Lam

Question #17266

Show that if a left ideal I ⊆ R is left quasi-regular, then it is quasi-regular.

Expert's answer

We must show that every

*b**∈**I*is*right*quasi-regular. Choose*a**∈**R*such that*a ◦ b*= 0. Then*a*=*ab− b**∈**I*so there exists*b'**∈**R*with*b' ◦ a*= 0. But then*b*= 0*◦b*= (*b' ◦ a*)*◦ b*=*b' ◦*(*a ◦ b*) =*b',*so we have*b ◦ a*= 0, as desired.
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