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Submodule Definition and Tensor Product

Tensor product of module homomorphismsOperation on a Tensor Product of ModulesShow that the image or the kernel are submodule of R-module.Tensor product over a tensor productDefinition of tensor productProperties of tensor product of modulesSimple submodule of modulesModule and submodule definitions, checking if I understood itSubmodule Criterion for Rings without 1If $Nsubset M$, where $M$ is an $R$-module and $rn_1 + n_2 in N$ for all $n_1,n_2 in N$ and $rin R$, then is $N$ a submodule?

0

$begingroup$

In Hungerford's Algebra, submodule is defined as follows:

Let $R$ be a ring, $A$ an $R$-module and $B$ a nonempty subset of $A$.
$B$ is a submodule of $A$ provided that $B$ is an additive subgroup of
$A$ and $rb in B$ for all $r in R$

Now, let $M$ and $N$ be $mathbbZ$-module and $H$ be a subset of $N$. Is it possible that $M otimes_mathbbZ H$ to be a submodule of $Motimes_mathbbZ N$ even if $H$ is not a subgroup of $N$ but $Motimes_mathbbZ H$ is additive subgroup of $Motimes_mathbbZ N$ and $rt in Motimes_mathbbZ H$ for all $rinmathbbZ$ and $t in Motimes_mathbbZ H$?

abstract-algebra ring-theory modules tensor-products

share|cite|improve this question

asked Mar 22 at 14:37

abuchayabuchay

363211

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What does $Motimes_mathbbZH$ mean if $H$ is not an abelian group?
  $endgroup$
  – Eric Wofsey
  Mar 22 at 19:13
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I believe $M$ and $H$ will fail to form a tensor product as a $mathbbZ$-module if $M$ is a $mathbbZ$-module and $H$ is just a subset and not an additive abelian group.
  $endgroup$
  – abuchay
  Mar 22 at 19:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What if $H$ is a multiplicative abelian group but not additive? For example, $H = 1$.
  $endgroup$
  – abuchay
  Mar 22 at 19:18
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Your question is completely meaningless until you say what you mean by $Motimes_mathbbZ H$.
  $endgroup$
  – Eric Wofsey
  Mar 22 at 19:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Note that even when $H$ is a submodule of $N$, $Motimes_mathbbZ H$ is not even a subset of $Motimes_mathbbZ N$ unless $H=N$. (There is a natural homomorphism $Motimes_mathbbZ Hto Motimes_mathbbZ N$ which is sometimes injective, and in those cases you can think of it as a submodule.)
  $endgroup$
  – Eric Wofsey
  Mar 22 at 19:21
  
  

  
  
  
  

 | 
show 1 more comment

0

$begingroup$

In Hungerford's Algebra, submodule is defined as follows:

Let $R$ be a ring, $A$ an $R$-module and $B$ a nonempty subset of $A$.
$B$ is a submodule of $A$ provided that $B$ is an additive subgroup of
$A$ and $rb in B$ for all $r in R$

Now, let $M$ and $N$ be $mathbbZ$-module and $H$ be a subset of $N$. Is it possible that $M otimes_mathbbZ H$ to be a submodule of $Motimes_mathbbZ N$ even if $H$ is not a subgroup of $N$ but $Motimes_mathbbZ H$ is additive subgroup of $Motimes_mathbbZ N$ and $rt in Motimes_mathbbZ H$ for all $rinmathbbZ$ and $t in Motimes_mathbbZ H$?

abstract-algebra ring-theory modules tensor-products

share|cite|improve this question

asked Mar 22 at 14:37

abuchayabuchay

363211

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What does $Motimes_mathbbZH$ mean if $H$ is not an abelian group?
  $endgroup$
  – Eric Wofsey
  Mar 22 at 19:13
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I believe $M$ and $H$ will fail to form a tensor product as a $mathbbZ$-module if $M$ is a $mathbbZ$-module and $H$ is just a subset and not an additive abelian group.
  $endgroup$
  – abuchay
  Mar 22 at 19:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What if $H$ is a multiplicative abelian group but not additive? For example, $H = 1$.
  $endgroup$
  – abuchay
  Mar 22 at 19:18
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Your question is completely meaningless until you say what you mean by $Motimes_mathbbZ H$.
  $endgroup$
  – Eric Wofsey
  Mar 22 at 19:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Note that even when $H$ is a submodule of $N$, $Motimes_mathbbZ H$ is not even a subset of $Motimes_mathbbZ N$ unless $H=N$. (There is a natural homomorphism $Motimes_mathbbZ Hto Motimes_mathbbZ N$ which is sometimes injective, and in those cases you can think of it as a submodule.)
  $endgroup$
  – Eric Wofsey
  Mar 22 at 19:21
  
  

  
  
  
  

 | 
show 1 more comment

$begingroup$

In Hungerford's Algebra, submodule is defined as follows:

Let $R$ be a ring, $A$ an $R$-module and $B$ a nonempty subset of $A$.
$B$ is a submodule of $A$ provided that $B$ is an additive subgroup of
$A$ and $rb in B$ for all $r in R$

Now, let $M$ and $N$ be $mathbbZ$-module and $H$ be a subset of $N$. Is it possible that $M otimes_mathbbZ H$ to be a submodule of $Motimes_mathbbZ N$ even if $H$ is not a subgroup of $N$ but $Motimes_mathbbZ H$ is additive subgroup of $Motimes_mathbbZ N$ and $rt in Motimes_mathbbZ H$ for all $rinmathbbZ$ and $t in Motimes_mathbbZ H$?

abstract-algebra ring-theory modules tensor-products

share|cite|improve this question

asked Mar 22 at 14:37

abuchayabuchay

363211

$endgroup$

share|cite|improve this question

asked Mar 22 at 14:37

abuchayabuchay

363211

share|cite|improve this question

asked Mar 22 at 14:37

abuchayabuchay

363211

asked Mar 22 at 14:37

abuchayabuchay

363211

asked Mar 22 at 14:37

abuchayabuchay

363211