Let R be a ring, MR a module, S a monoid, ω : S ?→ End(R)a monoid homomorphism and R ? S a skew monoid ring. Then M[S] ={m1g1 + · · · + mngn | n ≥ 1, mi ∈ Mand gi ∈ S for each 1 ≤ i ≤ n}is a module over R ? S. A module MR is Baer (resp. quasi-Baer ) if the annihilator of every subset (resp. submodule) of M is generated by an idempotent of R. In this paper we impose S-compatibility assumption on the module MR and prove: (1) MR is quasi-Baer if and only if M[s]RS is quasi-Baer, (2) MR is Baer (resp. p.p) if and only if M[S]RS is Baer (resp. p.p), where MR is S-skew Armendariz, (3) MR satisfies the ascending chain condition on annihilator of submodules if and only if so does M[S]R?S, where MR is S-skew quasi-Armendariz.
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