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Let G be a finite group, H ≤ G and R be a commutative ring with an identity 1_R. Let C_RG(H) = { α ∈ RG|αh= hα for all h ∈ H }, which is called the centralizer subalgebra of H in RG. Obviously, if H = G then C_RG(H) is just the central subalgebra Z(RG) of RG. In this note, we show that the set of all H-conjugacy class sums of G forms an R-basis of C_RG(H). Furthermore, let N be a normal subgroup of G and γ the natural epimorphism from G to G = G/N. Then γ induces an epimorphism from RG to RG, also denoted by γ. We also show that if R is a field of characteristic zero, then γ induces an epimorphism from C_RG(H) to C_RG (H), that is, γ(C_RG(H)) = C_RG (H). , number=3, journal=Communications in Mathematical Research, year=2021, month=May, pages=227–233