Definition:Additive Function on Class of Modules

Definition

Let $R$ be a commutative ring with unity.

Let $\CC$ be a class of $R$-modules.

Let $G$ be an abelian group.

Let $\lambda : \CC \to G$ be a mapping.

Then $\lambda$ is called an additive function if and only if:

for all $M' ', M, M' \in \CC$:

$0 \to M' ' \to M \to M' \to 0$ is a short exact sequence

$\implies \map \lambda {M' '} - \map \lambda M + \map \lambda {M'} = 0$

Sources

• 1969: M.F. Atiyah and I.G. MacDonald: Introduction to Commutative Algebra: Chapter $2$: Modules