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## Homework Statement

Let [itex]\mathbb{R}[/itex]*=[itex]\mathbb{R}[/itex]\{0} with multiplication operation. Show that [itex]\mathbb{R}[/itex]*=[itex]\mathbb{I}[/itex]

_{2}⊕ [itex]\mathbb{R}[/itex], where the group operation in [itex]\mathbb{R}[/itex] is addition.

## Homework Equations

Let {A

_{1},...,A

_{n}}[itex]\subseteq[/itex]A such that for all a[itex]\in[/itex]A there exists a unique sequence {a

_{k}} such that a=a

_{1}+...+a

_{n}where a

_{k}[itex]\in[/itex]A

_{k}for all k, then A=A

_{1}⊕...⊕A

_{n}

## The Attempt at a Solution

Since [itex]\mathbb{I}[/itex]

_{2}={-1,1} I don't think I can show that every a*[itex]\in[/itex][itex]\mathbb{R}[/itex]* can be expressed in a unique way. For example let a

^{+}=a*+1 and a

^{-}=a*-1, then a*=a

^{+}-1=a

^{-}+1. Am I defining the cyclic group of order 2 wrong? I'm not that sure about direct sums, our prof spent 5 minutes on them and 40% of our assignment involves them :S

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