Notes on Tensor Products

Basic Definition: Let R be a commutative ring with 1. A (unital) R-module is an abelian group M together with a operation R ×M → M , usually just written as rv when r ∈ R and v ∈ M . This operation is called scaling . The scaling operation satisfies the following conditions. 1. 1v = v for all v ∈M . 2. (rs)v = r(sv) for all r, s ∈ R and all v ∈M . 3. (r + s)v = rv + sv for all r, s ∈ R and all v ∈M . 4. r(v + w) = rv + rw for all r ∈ R and v, w ∈M . Technically, an R-module just satisfies properties 2, 3, 4. However, without the first property, the module is pretty pathological. So, we’ll always work with unital modules and just call them modules. When R is understood, we’ll just say module when we mean unital R-module.