View source for Exercises:Rings and Modules - 2016 - 1/Problem 1

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==Problems==
===Problem 1===
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====Part A====
Let {{M|R}} be a [[u-ring]]. Fix an {{M|a\in R}} and define a {{link|homomorphism|u-ring}}:
* {{M|\varphi_a:R[T]\rightarrow R}} by {{M|\varphi_a:P(T)\mapsto P(a)}} - evaluation at {{M|a}}.
By restriction of scalars every {{M|\varphi_a}} gives the target {{M|R}} the structure of an [[R-module|{{M|R[T]}}-module]], which we denote {{M|R_a}}.

Show that for {{M|a,b\in R}} that:
* there is an {{M|R[T]}}-module {{link|isomorphism|module}} between {{M|R_a}} and {{M|R_b}}
{{iff}}
* {{M|1=a=b}}
=====Solution=====
====Part B====
Let {{M|M}} be an {{M|R}}-module. Show that there is a [[surjection]] from a [[free module|free {{M|R}}-module]] ''onto'' {{M|M}}.
=====Solution=====
====Part C====
Show that the [[Z-Module|{{M|\mathbb{Z} }}-module]], {{M|\mathbb{Q} }}, is ''not'' free.
=====Solution=====
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==Notes==
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==References==
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