Exercises:Rings and Modules - 2016 - 1/Problem 1

Problems

Problem 1

Part A

Let [ilmath]R[/ilmath] be a u-ring. Fix an [ilmath]a\in R[/ilmath] and define a homomorphism:

• [ilmath]\varphi_a:R[T]\rightarrow R[/ilmath] by [ilmath]\varphi_a:P(T)\mapsto P(a)[/ilmath] - evaluation at [ilmath]a[/ilmath].

By restriction of scalars every [ilmath]\varphi_a[/ilmath] gives the target [ilmath]R[/ilmath] the structure of an [ilmath]R[T][/ilmath]-module, which we denote [ilmath]R_a[/ilmath].

Show that for [ilmath]a,b\in R[/ilmath] that:

• there is an [ilmath]R[T][/ilmath]-module isomorphism between [ilmath]R_a[/ilmath] and [ilmath]R_b[/ilmath]

if and only if

• [ilmath]a=b[/ilmath]

Part B

Let [ilmath]M[/ilmath] be an [ilmath]R[/ilmath]-module. Show that there is a surjection from a free [ilmath]R[/ilmath]-module onto [ilmath]M[/ilmath].

Part C

Show that the [ilmath]\mathbb{Z} [/ilmath]-module, [ilmath]\mathbb{Q} [/ilmath], is not free.