Thread: Modules

Let R be a commutative ring and let J be an Ideal in R. Recall That if M is an R-Module, then JM= { sum j_im_i } is a submodule of M. Prove that M/JM is an R/J module if we define scalar multiplication: (r+J)(m+JM) = rm + JM.

Conclude that If JM ={0}, then M itself is an R/J module. IF J is maximal ideal in R, then M is a vetor space over R/J.

If , then it is not hard to prove that . Now you can define what scalar multiplcation by elements of (R/J) to M meeans using this isomorphism.

Recall That if J is a maximal ideal of R, then R/J is a field. If M is an S-module, where S is a field, then M is an S-vector space.