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I am reading Joseph Rotman's book Advanced Modern Algebra.
I need help in fully understanding the proof of Proposition 1.52 on page 36.
Proposition 1.52 and its proof reads as follows:
View attachment 2676
The part of the proof on which I need help/clarification is Rotman's argument where he establishes that each $$ r \in U( \mathbb{I}_m ) $$ has an inverse in $$ U( \mathbb{I}_m ) $$.
As can be seen in the text above Rotman's argument (which I must say confuses me) reads as follows:
"If $$ (a,m) = 1 $$ then $$ [a][x] = 1 $$ can be solved for x in $$ \mathbb{I}_m $$. Now (x,m) = 1 for rx + sm = 1 for some integer s, and so (x,m) = 1. Hence $$ [x] \in U( \mathbb{I}_m ) $$, and so each $$ r \in U( \mathbb{I}_m ) $$ has an inverse in $$ U( \mathbb{I}_m ) $$."
I confess I cannot follow Rotman's argument above! (maybe MHB members will find it clearer than I do?)
Can someone please provide a clear and rigorous restatement of Rotman's argument regarding inverses in $$ U( \mathbb{I}_m ) $$.
Particular points of confusion are as follows:
1. Rotman writes: "If $$ (a,m) = 1 $$ then $$ [a][x] = 1 $$ can be solved for x in $$ \mathbb{I}_m $$." - I cannot see exactly why this follows:
2. Rotman writes: "Now (x,m) = 1 for rx + sm = 1 for some integer s, and so (x,m) = 1." - I do not follow this statement at all: indeed I think it may be badly expressed whatever he means ... ...
3. I do not follow the rest of his statements - probably because I do not follow 1 and 2 above.
Help and clarification would be appreciated
Peter
Note that Rotman uses the symbol $$\mathbb{I}_m$$ for $$ \mathbb{Z}/ m \mathbb{Z} $$.
I need help in fully understanding the proof of Proposition 1.52 on page 36.
Proposition 1.52 and its proof reads as follows:
View attachment 2676
The part of the proof on which I need help/clarification is Rotman's argument where he establishes that each $$ r \in U( \mathbb{I}_m ) $$ has an inverse in $$ U( \mathbb{I}_m ) $$.
As can be seen in the text above Rotman's argument (which I must say confuses me) reads as follows:
"If $$ (a,m) = 1 $$ then $$ [a][x] = 1 $$ can be solved for x in $$ \mathbb{I}_m $$. Now (x,m) = 1 for rx + sm = 1 for some integer s, and so (x,m) = 1. Hence $$ [x] \in U( \mathbb{I}_m ) $$, and so each $$ r \in U( \mathbb{I}_m ) $$ has an inverse in $$ U( \mathbb{I}_m ) $$."
I confess I cannot follow Rotman's argument above! (maybe MHB members will find it clearer than I do?)
Can someone please provide a clear and rigorous restatement of Rotman's argument regarding inverses in $$ U( \mathbb{I}_m ) $$.
Particular points of confusion are as follows:
1. Rotman writes: "If $$ (a,m) = 1 $$ then $$ [a][x] = 1 $$ can be solved for x in $$ \mathbb{I}_m $$." - I cannot see exactly why this follows:
2. Rotman writes: "Now (x,m) = 1 for rx + sm = 1 for some integer s, and so (x,m) = 1." - I do not follow this statement at all: indeed I think it may be badly expressed whatever he means ... ...
3. I do not follow the rest of his statements - probably because I do not follow 1 and 2 above.
Help and clarification would be appreciated
Peter
Note that Rotman uses the symbol $$\mathbb{I}_m$$ for $$ \mathbb{Z}/ m \mathbb{Z} $$.
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