Let m and n be left r-modules let f m n and g n m be left

Question

1. Let R be a ring. Prove that if x, y E R such that xy is right quasi-regular then yx is also right quasi-regular.

2. Let M and N be left R-modules. Let f : M N and g : N M be left R-module homomorphisms such that fg(y) = y for all y N. Show that M = ker(f) im(g).