[Module theory] Prove that something forms a left R module.

Your Name]In summary, to show that M\otimes N has the structure of a left S-module, you need to define a scalar multiplication that respects the relations of the tensor product. To do so, you can use the bilinearity and distributivity properties of the tensor product and the scalar multiplication in S. Let me know if you need further assistance.
  • #1
AdrianZ
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Homework Statement


Suppose that R and S are two rings, M, is a (R-S) bi-module and N is a left R-module. Show that [itex] M \otimes N [/itex] has the structure of a left S-module.

The Attempt at a Solution



Well, [itex] M\otimes N [/itex] is an Abelian group, so it's enough that I define a scalar product on [itex] M\otimes N [/itex]. I'm thinking of defining:
[itex] s.\sum_{i=1}^t{x_i\otimes y_i} = \sum_{i=1}^t{sx_i\otimes y_i} [/itex]
Now I'm a bit clueless about how I should show that this scalar multiplication is well-defined. I know that I should suppose [itex] \sum_{i=1}^t{x_i\otimes y_i} = \sum_{i=1}^{t'}{x'_i\otimes y'_i} [/itex] and then show that [itex] \sum_{i=1}^t{sx_i\otimes y_i} = \sum_{i=1}^{t'}{sx'_i\otimes y'_i} [/itex] but I don't know how I should do that. I'm looking for a nice map from [itex]F[/itex] to [itex]F/K = M\otimes N[/itex] that does the trick but nothing good comes to my mind now. Any ideas would be appreciated greatly.
Thanks in advance.
 
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  • #2


Thank you for your question. It seems like you have the right idea about defining a scalar product on M\otimes N to show that it has the structure of a left S-module. To show that this scalar multiplication is well-defined, you can use the universal property of tensor products.

Recall that the tensor product M\otimes N is defined as the quotient of the free abelian group on the set M\times N by the subgroup generated by elements of the form:

1) (x+x',y) - (x,y) - (x',y)
2) (x,y+y') - (x,y) - (x,y')
3) (rx,y) - r(x,y)
4) (x,ry) - r(x,y)

where x,x' \in M, y,y' \in N and r \in R. So, to show that your scalar multiplication is well-defined, you need to show that it respects these relations. For example, to show that it respects relation 1), you can use the bilinearity of the tensor product and the distributivity of the scalar multiplication in S:

s.\left((x+x')\otimes y\right) = s.\left(x\otimes y + x'\otimes y\right) = s.\left(x\otimes y\right) + s.\left(x'\otimes y\right)

Similarly, you can show that your scalar multiplication respects the other relations as well. I hope this helps. Let me know if you need any further clarification or assistance.
 

1. How do you define a left R module?

A left R module is a mathematical structure consisting of a ring R and an abelian group M, where R acts on M through left multiplication, satisfying certain properties such as associativity, distributivity, and compatibility with the group structure of M.

2. What is the significance of proving that something forms a left R module?

Proving that something forms a left R module is important in understanding the structure and properties of the object in question. It also allows for the application of various techniques and theorems from module theory, which is a fundamental area of study in abstract algebra.

3. What are the steps involved in proving that something forms a left R module?

The steps involved in proving that something forms a left R module typically include defining the ring R, the abelian group M, and the action of R on M, and then verifying the module axioms, which include the properties of associativity, distributivity, and compatibility mentioned earlier.

4. How does one show that the module axioms are satisfied?

To show that the module axioms are satisfied, one must demonstrate that the given set M, together with the specified ring R and action of R on M, satisfies all the required properties. This can be done by using known properties of R and M, and by applying the definitions of the module axioms.

5. Can something form both a left and right R module?

Yes, it is possible for something to form both a left and right R module. This is known as a bimodule and has properties that are a combination of those for left and right modules. Proving that something is a bimodule involves showing that it satisfies both the left and right module axioms.

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