On \({\mathbb{Z}}\)-valued additive functions on module category
Let R be a commutative ring with an identity and \({\mathcal C}(R)\) the category of unitary R-modules. For \((S,+)\), a set with binary operation \(+\), \(L:\quad {\mathcal C}(R)\to S\) is called an S-valued additive function over R if for any exact sequence \(0\to M'\to M\to M''\to 0\) in \({\mathcal C}(R)\), \(L(M)=L(M')+L(M'').\) The study begins with R a noetherian ring, and additive functions on the quotient of R by an ideal, which then can be extended to R[X], the polynomial ring in one variable. The author then considers the case \(S={\mathbb{Z}}\), and studies the \({\mathbb{Z}}\)-valued additive functions. The paper ends mentioning a rather nice result of S. Kondo on non-trivial additive functions.
Grothendieck group of the category of modules
scientific article; zbMATH DE number 3943985