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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