Answer to Question #17266 in Abstract Algebra for Tsit Lam

Question

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

Answer 1

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.