Question

Show that if ab is left quasi-regular element of ring , then so is ba.

Answer 1

Question 1. Show that if $ab$ is left quasi-regular element of ring, then so is $ba$.

Solution. Suppose $ab$ is quasi-regular, so there is $c$ such that $0 = c \circ ab = ab \circ c$. Consider $d = b(c - 1)a$. Note that $d \circ ba = b(c - 1)a + ba - b(c - 1)aba = bca - ba + ba - bcaba + baba = b(c - cab + ab)a = b(c \circ ab)a = b \cdot 0 \cdot a = 0$. And similarly $ba \circ d = ba + b(c - 1)a - bab(c - 1)a = ba + bca - ba - babca + baba = b(c - abc + ab)a = b(ab \circ c)a = b \cdot 0 \cdot a = 0$. Thus, $ba$ is quasi-regular.