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

Problems

Problem 1

Part A

Let $R$ be a u-ring. Fix an $a\in R$ and define a homomorphism:

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

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

Show that for $a,b\in R$ that:

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

if and only if

• $a=b$

Part B

Let $M$ be an $R$-module. Show that there is a surjection from a free $R$-module onto $M$.

Part C

Show that the $\mathbb{Z}$-module, $\mathbb{Q}$, is not free.