Proof. Consider the map g:A/I → C , a+I 7→ f (a). It is well defined: a+I = a′ +I implies a− a′ ∈ I implies f (a) = f (a′). The element a + I belongs to the kernel of g iff g(a + I) = f (a) = 0, i.e. a ∈ I , i.e. a + I = I is the zero element of A/I . Thus, ker(g) = 0. The image of g is g(A/I) = {f (a) : a ∈ A} = C . Thus, g is an isomorphism. The inverse morphism to g is given by f (a) 7→ a + I .

In this paper, we introduce and investigate the notion of projection invariant semisimple modules. Some structural properties aforementioned class modules are studied. We obtain indecomposable decompositions former under some module theoretical conditions. Moreover, explore when finite exchange property implies full for Finally, that endomorphism ring a is ?- Baer ring.

1 Definitions Definition 1. A graded ring is a ring S together with a set of subgroups Sd, d ≥ 0 such that S = ⊕ d≥0 Sd as an abelian group, and st ∈ Sd+e for all s ∈ Sd, t ∈ Se. One can prove that 1 ∈ S0 and if S is a domain then any unit of S also belongs to S0. A homogenous ideal of S is an ideal a with the property that for any f ∈ a we also have fd ∈ a for all d ≥ 0. A morphism of graded r...

Many known results on finite von Neumann algebras are generalized, by purely algebraic proofs, to a certain class C of finite Baer *-rings. The results in this paper can also be viewed as a study of the properties of Baer *-rings in the class C. First, we show that a finitely generated module over a ring from the class C splits as a direct sum of a finitely generated projective module and a cer...