How Do We Prove Something is a Submodule?

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In summary, the conversation discusses the criteria for proving that a subset is a submodule of a left R-module. These criteria include being closed under scalar multiplication, containing the additive identity, being closed under addition, and containing the additive inverse of any element. While the linked proof only shows the first and third criteria, the others can be easily derived from the first criterion by setting r = 0 and r = -1.
  • #1
Artusartos
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My textbook says that...

If M is a left R-module, then a submodule N of M...is an additive subgoup N of M closed under scalar multiplication: [tex]rn \in N[/tex] whenever [tex]n \in N[/tex] and [tex]r \in R[/tex].

So if we want to prove that something is a submodule, we need to show that...

1) It closed under scalar multiplication
2) The additive idenitity is in N
3) N is closed under additition
4) If x is in N, then so is its inverse

Right?

But, in the link that I attached, it only shows 1) and 3), right? Can anybody tell me why? Is the proof still considered complete?

Thanks in advance
 

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  • #2
Isn't it as simple as: If (1) holds, set r = 0 to get (2) and r = -1 to get (4)?
(OK, you might want to show that 0n is the additive identity for any n and that -n is the additive inverse of any n).
 
  • #3
CompuChip said:
Isn't it as simple as: If (1) holds, set r = 0 to get (2) and r = -1 to get (4)?
(OK, you might want to show that 0n is the additive identity for any n and that -n is the additive inverse of any n).

Thanks.
 
  • #4
1 does not imply 2, unless the subset considered is non empty. i.e. 1 implies that IF the subset contains anything, then it also contains 0.
 
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  • #5


I would like to clarify that the proof provided in the link is indeed complete. However, it may not explicitly mention all four conditions because they can be inferred from the given information.

The first condition, closure under scalar multiplication, is shown by the statement "rn \in N whenever n \in N and r \in R". This implies that N is closed under scalar multiplication.

The second condition, the additive identity being in N, is implied by the fact that N is a subgroup of M. This means that it must contain the identity element of M, which is also the additive identity.

The third condition, closure under addition, is shown by the statement "x+y \in N whenever x,y \in N". This implies that N is closed under addition.

The fourth condition, closure under inverses, is not explicitly mentioned because it is not necessary for a submodule to contain the inverse of every element. As long as N is closed under addition and scalar multiplication, it is considered a submodule. The inverse of an element can always be obtained by multiplying it with -1.

In conclusion, the proof provided in the link is complete and the four conditions for a submodule are implicitly shown. However, it is always important to be aware of all the necessary conditions and make sure they are met when proving something is a submodule.
 

FAQ: How Do We Prove Something is a Submodule?

What is a submodule?

A submodule is a subset of a larger module, similar to a folder within a directory. It allows for organization and compartmentalization within a larger codebase.

How are submodules used in scientific research?

Submodules can be used in scientific research to break down complex projects into smaller, more manageable components. They can also be used to incorporate code or data from other sources into a research project.

What are the benefits of using submodules?

Using submodules can improve the efficiency and organization of a project. They also allow for easier collaboration and sharing of code among team members. Additionally, submodules can help with version control and tracking changes to specific components of a project.

Can submodules be nested within other submodules?

Yes, submodules can be nested within other submodules. This can be useful for organizing and structuring complex projects with multiple layers.

Are there any potential challenges or drawbacks to using submodules?

One potential challenge of using submodules is ensuring that all parts of the project are properly connected and functioning together. Additionally, if not managed properly, submodules can lead to version control issues and conflicts. It is important to carefully plan and maintain submodules within a project.

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