Isomorphism of Tensor Products












1












$begingroup$


$M bigotimes_{mathbb{Z}} mathbb{Z}_{21} cong (M/3M) bigoplus (M/7M)$



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







share|cite|improve this question











$endgroup$












  • $begingroup$
    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


















1












$begingroup$


$M bigotimes_{mathbb{Z}} mathbb{Z}_{21} cong (M/3M) bigoplus (M/7M)$



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







share|cite|improve this question











$endgroup$












  • $begingroup$
    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
















1












1








1





$begingroup$


$M bigotimes_{mathbb{Z}} mathbb{Z}_{21} cong (M/3M) bigoplus (M/7M)$



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







share|cite|improve this question











$endgroup$




$M bigotimes_{mathbb{Z}} mathbb{Z}_{21} cong (M/3M) bigoplus (M/7M)$



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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 2 at 11:58







Pedro Santos

















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




















  • $begingroup$
    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












1 Answer
1






active

oldest

votes


















3












$begingroup$

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?






share|cite|improve this answer











$endgroup$













  • $begingroup$
    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











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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









3












$begingroup$

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?






share|cite|improve this answer











$endgroup$













  • $begingroup$
    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
















3












$begingroup$

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?






share|cite|improve this answer











$endgroup$













  • $begingroup$
    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














3












3








3





$begingroup$

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?






share|cite|improve this answer











$endgroup$



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?







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Jan 2 at 12:02

























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


















  • $begingroup$
    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


















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