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

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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
Solution

Part B

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

Solution

Part C

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

Solution

Notes

References