Paper 4, Section II, G
Part IB, 2012
Let be a commutative ring with unit 1. Prove that an -module is finitely generated if and only if it is a quotient of a free module , for some .
Let be a finitely generated -module. Suppose now is an ideal of , and is an -homomorphism from to with the property that
Prove that satisfies an equation
where each . [You may assume that if is a matrix over , then (id), with id the identity matrix.]
Deduce that if satisfies , then there is some satisfying
Give an example of a finitely generated -module and a proper ideal of satisfying the hypothesis , and for your example, give an explicit such element .