Proof. Let A ∈ L(R). Since M is faithful and weak-join-quasi-cyclic, we have (AB : B) =
(ABM : BM). As BM is weak-join-quasi-cyclic, we have (ABM : BM) = A+(0 : BM) = A+((0 :
M) : B) = A+(0 : B). Therefore B is weak join principal.
Let A, C ∈ L(R). Since M is faithful and weak-join-quasi-cyclic, it follows that ((AB + C) :
B) = ((ABM + CM) : BM). As BM is join-quasi-cyclic, we have ((ABM + CM) : BM) =
A+ (CM : BM) = A+ (C : B) since M is faithful and weak-join-quasi-cyclic. Therefore B is
join principal.