Isomorphism of Tensor Products
$begingroup$
$M bigotimes_{mathbb{Z}} mathbb{Z}_{21} cong (M/3M) bigoplus (M/7M)$
Can u guys help me prove if this is true or not? M is an abelian group, so a $mathbb{Z}$ module I tried creating homomorphisms with the universal property of the tensor product for bilinear maps and prove that they are inverse to one another but im not getting the desire isomorphism, yet i dont know if the statement is true.
abstract-algebra modules tensor-products
asked Jan 2 at 11:51
Pedro SantosPedro Santos
1459
$endgroup$
add a comment |
$begingroup$
$M bigotimes_{mathbb{Z}} mathbb{Z}_{21} cong (M/3M) bigoplus (M/7M)$
Can u guys help me prove if this is true or not? M is an abelian group, so a $mathbb{Z}$ module I tried creating homomorphisms with the universal property of the tensor product for bilinear maps and prove that they are inverse to one another but im not getting the desire isomorphism, yet i dont know if the statement is true.
abstract-algebra modules tensor-products
asked Jan 2 at 11:51
Pedro SantosPedro Santos
1459
$endgroup$
$begingroup$
what have you tried so far?
$endgroup$
– Praphulla Koushik
Jan 2 at 11:52
$begingroup$
ive tried creating maps , for example one that sends $(m+3M,m_1+7M) to (mm_1 bigotimes 1 +21mathbb{Z})$ and one that sends $ (m bigotimes x+21mathbb{Z}) to (xm + 3M, 1 +7M)$
$endgroup$
– Pedro Santos
Jan 2 at 11:56
add a comment |
$begingroup$
$M bigotimes_{mathbb{Z}} mathbb{Z}_{21} cong (M/3M) bigoplus (M/7M)$
Can u guys help me prove if this is true or not? M is an abelian group, so a $mathbb{Z}$ module I tried creating homomorphisms with the universal property of the tensor product for bilinear maps and prove that they are inverse to one another but im not getting the desire isomorphism, yet i dont know if the statement is true.
abstract-algebra modules tensor-products
asked Jan 2 at 11:51
Pedro SantosPedro Santos
1459
$endgroup$
$M bigotimes_{mathbb{Z}} mathbb{Z}_{21} cong (M/3M) bigoplus (M/7M)$
Can u guys help me prove if this is true or not? M is an abelian group, so a $mathbb{Z}$ module I tried creating homomorphisms with the universal property of the tensor product for bilinear maps and prove that they are inverse to one another but im not getting the desire isomorphism, yet i dont know if the statement is true.
abstract-algebra modules tensor-products
abstract-algebra modules tensor-products
asked Jan 2 at 11:51
Pedro SantosPedro Santos
1459
asked Jan 2 at 11:51
Pedro SantosPedro Santos
1459
edited Jan 2 at 11:58
Pedro Santos
asked Jan 2 at 11:51
Pedro SantosPedro Santos
1459
asked Jan 2 at 11:51
Pedro SantosPedro Santos
1459
asked Jan 2 at 11:51
Pedro SantosPedro Santos
1459
1459
$begingroup$
what have you tried so far?
$endgroup$
– Praphulla Koushik
Jan 2 at 11:52
$begingroup$
ive tried creating maps , for example one that sends $(m+3M,m_1+7M) to (mm_1 bigotimes 1 +21mathbb{Z})$ and one that sends $ (m bigotimes x+21mathbb{Z}) to (xm + 3M, 1 +7M)$
$endgroup$
– Pedro Santos
Jan 2 at 11:56
add a comment |
$begingroup$
what have you tried so far?
$endgroup$
– Praphulla Koushik
Jan 2 at 11:52
$begingroup$
ive tried creating maps , for example one that sends $(m+3M,m_1+7M) to (mm_1 bigotimes 1 +21mathbb{Z})$ and one that sends $ (m bigotimes x+21mathbb{Z}) to (xm + 3M, 1 +7M)$
$endgroup$
– Pedro Santos
Jan 2 at 11:56
$begingroup$
what have you tried so far?
$endgroup$
– Praphulla Koushik
Jan 2 at 11:52
$begingroup$
what have you tried so far?
$endgroup$
– Praphulla Koushik
Jan 2 at 11:52
$begingroup$
ive tried creating maps , for example one that sends $(m+3M,m_1+7M) to (mm_1 bigotimes 1 +21mathbb{Z})$ and one that sends $ (m bigotimes x+21mathbb{Z}) to (xm + 3M, 1 +7M)$
$endgroup$
– Pedro Santos
Jan 2 at 11:56
$begingroup$
ive tried creating maps , for example one that sends $(m+3M,m_1+7M) to (mm_1 bigotimes 1 +21mathbb{Z})$ and one that sends $ (m bigotimes x+21mathbb{Z}) to (xm + 3M, 1 +7M)$
$endgroup$
– Pedro Santos
Jan 2 at 11:56
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
See that $mathbb{Z}_{21}=mathbb{Z}_3oplus mathbb{Z}_7$.
As tensor product distributes with direct sum, $$Motimes_{mathbb{Z}}mathbb{Z}_{21}=Motimes_{mathbb{Z}}(mathbb{Z}_3oplus mathbb{Z}_7)=(Motimes_mathbb{Z}mathbb{Z}_3)oplus (Motimes_mathbb{Z}mathbb{Z}_7)$$
Can you take it from here?
answered Jan 2 at 11:57
Praphulla KoushikPraphulla Koushik
16919
$endgroup$
$begingroup$
Oh yeah i forgot about that damn it , Thanks , now i can use the fact that $M bigotimes mathbb{Z}{m} cong M/mM$
$endgroup$
– Pedro Santos
Jan 2 at 11:59
$begingroup$
Enjoy!!!!!!!!!!
$endgroup$
– Praphulla Koushik
Jan 2 at 12:01
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
See that $mathbb{Z}_{21}=mathbb{Z}_3oplus mathbb{Z}_7$.
As tensor product distributes with direct sum, $$Motimes_{mathbb{Z}}mathbb{Z}_{21}=Motimes_{mathbb{Z}}(mathbb{Z}_3oplus mathbb{Z}_7)=(Motimes_mathbb{Z}mathbb{Z}_3)oplus (Motimes_mathbb{Z}mathbb{Z}_7)$$
Can you take it from here?
answered Jan 2 at 11:57
Praphulla KoushikPraphulla Koushik
16919
$endgroup$
$begingroup$
Oh yeah i forgot about that damn it , Thanks , now i can use the fact that $M bigotimes mathbb{Z}{m} cong M/mM$
$endgroup$
– Pedro Santos
Jan 2 at 11:59
$begingroup$
Enjoy!!!!!!!!!!
$endgroup$
– Praphulla Koushik
Jan 2 at 12:01
add a comment |
$begingroup$
See that $mathbb{Z}_{21}=mathbb{Z}_3oplus mathbb{Z}_7$.
As tensor product distributes with direct sum, $$Motimes_{mathbb{Z}}mathbb{Z}_{21}=Motimes_{mathbb{Z}}(mathbb{Z}_3oplus mathbb{Z}_7)=(Motimes_mathbb{Z}mathbb{Z}_3)oplus (Motimes_mathbb{Z}mathbb{Z}_7)$$
Can you take it from here?
answered Jan 2 at 11:57
Praphulla KoushikPraphulla Koushik
16919
$endgroup$
$begingroup$
Oh yeah i forgot about that damn it , Thanks , now i can use the fact that $M bigotimes mathbb{Z}{m} cong M/mM$
$endgroup$
– Pedro Santos
Jan 2 at 11:59
$begingroup$
Enjoy!!!!!!!!!!
$endgroup$
– Praphulla Koushik
Jan 2 at 12:01
add a comment |
$begingroup$
See that $mathbb{Z}_{21}=mathbb{Z}_3oplus mathbb{Z}_7$.
As tensor product distributes with direct sum, $$Motimes_{mathbb{Z}}mathbb{Z}_{21}=Motimes_{mathbb{Z}}(mathbb{Z}_3oplus mathbb{Z}_7)=(Motimes_mathbb{Z}mathbb{Z}_3)oplus (Motimes_mathbb{Z}mathbb{Z}_7)$$
Can you take it from here?
answered Jan 2 at 11:57
Praphulla KoushikPraphulla Koushik
16919
$endgroup$
See that $mathbb{Z}_{21}=mathbb{Z}_3oplus mathbb{Z}_7$.
As tensor product distributes with direct sum, $$Motimes_{mathbb{Z}}mathbb{Z}_{21}=Motimes_{mathbb{Z}}(mathbb{Z}_3oplus mathbb{Z}_7)=(Motimes_mathbb{Z}mathbb{Z}_3)oplus (Motimes_mathbb{Z}mathbb{Z}_7)$$
Can you take it from here?
answered Jan 2 at 11:57
Praphulla KoushikPraphulla Koushik
16919
edited Jan 2 at 12:02
answered Jan 2 at 11:57
Praphulla KoushikPraphulla Koushik
16919
answered Jan 2 at 11:57
Praphulla KoushikPraphulla Koushik
16919
answered Jan 2 at 11:57
Praphulla KoushikPraphulla Koushik
16919
16919
$begingroup$
Oh yeah i forgot about that damn it , Thanks , now i can use the fact that $M bigotimes mathbb{Z}{m} cong M/mM$
$endgroup$
– Pedro Santos
Jan 2 at 11:59
$begingroup$
Enjoy!!!!!!!!!!
$endgroup$
– Praphulla Koushik
Jan 2 at 12:01
add a comment |
$begingroup$
Oh yeah i forgot about that damn it , Thanks , now i can use the fact that $M bigotimes mathbb{Z}{m} cong M/mM$
$endgroup$
– Pedro Santos
Jan 2 at 11:59
$begingroup$
Enjoy!!!!!!!!!!
$endgroup$
– Praphulla Koushik
Jan 2 at 12:01
$begingroup$
Oh yeah i forgot about that damn it , Thanks , now i can use the fact that $M bigotimes mathbb{Z}{m} cong M/mM$
$endgroup$
– Pedro Santos
Jan 2 at 11:59
$begingroup$
Oh yeah i forgot about that damn it , Thanks , now i can use the fact that $M bigotimes mathbb{Z}{m} cong M/mM$
$endgroup$
– Pedro Santos
Jan 2 at 11:59
$begingroup$
Enjoy!!!!!!!!!!
$endgroup$
– Praphulla Koushik
Jan 2 at 12:01
$begingroup$
Enjoy!!!!!!!!!!
$endgroup$
– Praphulla Koushik
Jan 2 at 12:01
add a comment |
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$begingroup$
what have you tried so far?
$endgroup$
– Praphulla Koushik
Jan 2 at 11:52
$begingroup$
ive tried creating maps , for example one that sends $(m+3M,m_1+7M) to (mm_1 bigotimes 1 +21mathbb{Z})$ and one that sends $ (m bigotimes x+21mathbb{Z}) to (xm + 3M, 1 +7M)$
$endgroup$
– Pedro Santos
Jan 2 at 11:56