Paper 4, Section II, G

Groups, Rings and Modules
Part IB, 2012

Let RR be a commutative ring with unit 1. Prove that an RR-module is finitely generated if and only if it is a quotient of a free module RnR^{n}, for some n>0n>0.

Let MM be a finitely generated RR-module. Suppose now II is an ideal of RR, and ϕ\phi is an RR-homomorphism from MM to MM with the property that

ϕ(M)IM={mMm=rm with rI,mM}\phi(M) \subset I \cdot M=\left\{m \in M \mid m=r m^{\prime} \quad \text { with } \quad r \in I, m^{\prime} \in M\right\}

Prove that ϕ\phi satisfies an equation

ϕn+an1ϕn1++a1ϕ+a0=0\phi^{n}+a_{n-1} \phi^{n-1}+\cdots+a_{1} \phi+a_{0}=0

where each ajIa_{j} \in I. [You may assume that if TT is a matrix over RR, then adj(T)T=\operatorname{adj}(T) T= detT\operatorname{det} T (id), with id the identity matrix.]

Deduce that if MM satisfies IM=MI \cdot M=M, then there is some aRa \in R satisfying

a1I and aM=0.a-1 \in I \quad \text { and } \quad a M=0 .

Give an example of a finitely generated Z\mathbb{Z}-module MM and a proper ideal II of Z\mathbb{Z} satisfying the hypothesis IM=MI \cdot M=M, and for your example, give an explicit such element aa.