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