Abstract
Problem statement: Algebraic K-theory of projective modules over commutative rings were introduced by Bass and central simple superalgebras, supercommutative super-rings were introduced by many researchers such as Knus, Racine and Zelmanov. In this research, we classified the projective supermodules over (torsion free) supercommutative super-rings and through out this study we forced our selves to generalize the algebraic K-theory of projective supermodules over (torsion free) supercommutative super-rings. Approach: We generalized the algebraic K-theory of projective modules to the super-case over (torsion free) supercommutative super-rings. Results: we extended two results proved by Saltman to the supercase. Conclusion: The extending two results, which were proved by Saltman, to the supercase and the algebraic K-theory of projective supermodules over (torsion free) supercommutative super-rings would help any researcher to classify further properties about projective supermodules.
Highlights
An associative super-ring R = R0 + R1 is nothing but a Z2 -graded associative ring
In[1], we generalized the same results about finitely generated projective supermodules over R, where R is a supercommutative super-ring
In the theorem we try to find the conditions on A= EndR(P) to have a superinvolution of any kind, where P is an R-progenerator as a supermodule over R, if R is a connected super-ring
Summary
In[1], we generalized the same results about finitely generated projective supermodules over R, where R is a supercommutative super-ring. The Grothendieck group of C is defined to be an abelian group K0 C, together with a map: Suppose C is any category and obj(C) the class of all objects of C and let C(A,B) be the set of all morphisms A→B, where A,B ∈obj(C). P(R) denote the category of finitely generated projective supermodules over R with isomorphisms
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