On extensions of Baer and quasi-Baer modules

Let R be a ring, M_R a module, S a monoid, ω : S → End(R) a monoid homomorphism and R * S a skew monoid ring. Then M[S] = {m₁g₁ + · · · + m_ng_n | n ≥ 1, m_i ∈ M and g_i ∈ S for each 1 ≤ i ≤ n} is a module over R ∗ S. A module M_R 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 M_R and prove: (1) M_R is quasi-Baer if and only if M[s]_R∗S is quasi-Baer, (2) M_R is Baer (resp. p.p) if and only if M[S]_R∗S is Baer (resp. p.p), where M_R is S-skew Armendariz, (3) M_R satisfies the ascending chain condition on annihilator of submodules if and only if so does M[S]_R∗S, where M_R is S-skew quasi-Armendariz.