Invariant factors of a module over a $PID$
$begingroup$
so i need some help. Imagine D is Principal Ideal Domain, and $M$ and $N$ are cyclic modules with order $a$ and $b$ respectevly, and $mdc(a,b)neq 1$. So im asked to prove that the invariant factors of $Mbigoplus N$ are the $gcd(a,b)$ and the $lcm(a,b)$. Im not quite sure how to do this, i know that $M cong D/(a)$ and $N cong D/(b)$ but i dont know what to do from here.Am i supposed to try create isomorphisms to prove that those invariant factors work? Any help is appreciated, Thanks.
abstract-algebra modules principal-ideal-domains
asked Dec 30 '18 at 10:35
Pedro SantosPedro Santos
817
$endgroup$
add a comment |
$begingroup$
so i need some help. Imagine D is Principal Ideal Domain, and $M$ and $N$ are cyclic modules with order $a$ and $b$ respectevly, and $mdc(a,b)neq 1$. So im asked to prove that the invariant factors of $Mbigoplus N$ are the $gcd(a,b)$ and the $lcm(a,b)$. Im not quite sure how to do this, i know that $M cong D/(a)$ and $N cong D/(b)$ but i dont know what to do from here.Am i supposed to try create isomorphisms to prove that those invariant factors work? Any help is appreciated, Thanks.
abstract-algebra modules principal-ideal-domains
asked Dec 30 '18 at 10:35
Pedro SantosPedro Santos
817
$endgroup$
1
$begingroup$
See math.stackexchange.com/a/2226275/589
$endgroup$
– lhf
Dec 30 '18 at 10:51
$begingroup$
Alright so i do have to create the isomorphism , thx !!! I was just wondering if there was another way to do it.
$endgroup$
– Pedro Santos
Dec 30 '18 at 10:54
add a comment |
$begingroup$
so i need some help. Imagine D is Principal Ideal Domain, and $M$ and $N$ are cyclic modules with order $a$ and $b$ respectevly, and $mdc(a,b)neq 1$. So im asked to prove that the invariant factors of $Mbigoplus N$ are the $gcd(a,b)$ and the $lcm(a,b)$. Im not quite sure how to do this, i know that $M cong D/(a)$ and $N cong D/(b)$ but i dont know what to do from here.Am i supposed to try create isomorphisms to prove that those invariant factors work? Any help is appreciated, Thanks.
abstract-algebra modules principal-ideal-domains
asked Dec 30 '18 at 10:35
Pedro SantosPedro Santos
817
$endgroup$
so i need some help. Imagine D is Principal Ideal Domain, and $M$ and $N$ are cyclic modules with order $a$ and $b$ respectevly, and $mdc(a,b)neq 1$. So im asked to prove that the invariant factors of $Mbigoplus N$ are the $gcd(a,b)$ and the $lcm(a,b)$. Im not quite sure how to do this, i know that $M cong D/(a)$ and $N cong D/(b)$ but i dont know what to do from here.Am i supposed to try create isomorphisms to prove that those invariant factors work? Any help is appreciated, Thanks.
abstract-algebra modules principal-ideal-domains
abstract-algebra modules principal-ideal-domains
asked Dec 30 '18 at 10:35
Pedro SantosPedro Santos
817
asked Dec 30 '18 at 10:35
Pedro SantosPedro Santos
817
asked Dec 30 '18 at 10:35
Pedro SantosPedro Santos
817
asked Dec 30 '18 at 10:35
Pedro SantosPedro Santos
817
asked Dec 30 '18 at 10:35
Pedro SantosPedro Santos
817
817
1
$begingroup$
See math.stackexchange.com/a/2226275/589
$endgroup$
– lhf
Dec 30 '18 at 10:51
$begingroup$
Alright so i do have to create the isomorphism , thx !!! I was just wondering if there was another way to do it.
$endgroup$
– Pedro Santos
Dec 30 '18 at 10:54
add a comment |
1
$begingroup$
See math.stackexchange.com/a/2226275/589
$endgroup$
– lhf
Dec 30 '18 at 10:51
$begingroup$
Alright so i do have to create the isomorphism , thx !!! I was just wondering if there was another way to do it.
$endgroup$
– Pedro Santos
Dec 30 '18 at 10:54
1
1
$begingroup$
See math.stackexchange.com/a/2226275/589
$endgroup$
– lhf
Dec 30 '18 at 10:51
$begingroup$
See math.stackexchange.com/a/2226275/589
$endgroup$
– lhf
Dec 30 '18 at 10:51
$begingroup$
Alright so i do have to create the isomorphism , thx !!! I was just wondering if there was another way to do it.
$endgroup$
– Pedro Santos
Dec 30 '18 at 10:54
$begingroup$
Alright so i do have to create the isomorphism , thx !!! I was just wondering if there was another way to do it.
$endgroup$
– Pedro Santos
Dec 30 '18 at 10:54
add a comment |
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1
$begingroup$
See math.stackexchange.com/a/2226275/589
$endgroup$
– lhf
Dec 30 '18 at 10:51
$begingroup$
Alright so i do have to create the isomorphism , thx !!! I was just wondering if there was another way to do it.
$endgroup$
– Pedro Santos
Dec 30 '18 at 10:54