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In this paper, we introduce the concepts of semi-Baer, semi-quasi Baer, semi-p.q. Baer and semi-p.p. modules as a generalization of Baer, quasi Baer, p.q. Baer and p.p. modules respectively. To define these concepts, we introduce concepts of multiplicative order of an element and a multiplicatively finite element in rings. Further, we characterize these concepts in modules over reduced rings. Also, it is proved that semi-Baer and semi-quasi Baer properties are preserved by polynomial extensions and power series extensions of modules. It is proved that for a ring R and a monoid G, if the semi group ring RG is semi-Baer (semi-quasi Baer) then so is R.

We prove that if a group ring RG is a (quasi) Baer *-ring, then so is R, whereas converse is not true. Sufficient conditions are given so that for some finite cyclic groups G, if R is (quasi-) Baer *-ring, then so is the group ring RG. We prove that if the group ring RG is a Baer *-ring, then so is RH for every subgroup H of G. Also, we generalize results of Zhong Yi, Yiqiang Zhou (for (quasi-) Baer rings) and L. Zan, J. Chen (for principally quasi-Baer and principally projective rings).

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