# On a finiteley generated submodule of a direct sum of left R-modules

• MHB
• steenis
In summary, each $x_j$ is in a finite direct sum of $M_i$'s. This is confirmed by the fact that each $x_j$ is an element of $M$, which is an external direct sum of left $R$-modules.
steenis
Suppose $(M_i)_{i \in I}$ is a family of left $R$-modules and $M = \bigoplus_{i \in I} M_i$.
Suppose $N = \langle x_1 \cdots x_m \rangle$ is a finitely generated submodule of $M$.
Then for each $j = 1 \cdots m$, there is a finite $I_j \subset I$ such that $x_j \in \bigoplus_{i \in I_j} M_i$.

Can anyone help me with this ?*EDIT*
It is clear that each $x_j$ is in at least one $M_i$. I even think that it is exactly one. I need some confirmation.$Last edited: My *EDIT* in post #1 in not correct.It took me a while, but the solution is: Each$x_j$is an element of$M = \bigoplus_{ i \in I} M_i $. Therefore$x_j = (m_{ij})_{I \in I}$where$m_{ij} \in M_i$is the i-th component of$x_j$in$\bigoplus_{ i \in I} M_i $. M is an external direct sum, so only finitely many$m_{ij}$are nonzero. Let$I_j = \{i \in I | m_{ij} \neq 0 \}$,$I_j$is finite. Then$x_j \in \bigoplus_{ i \in I_j} M_i \$.

Last edited:

## 1. What is a submodule?

A submodule is a subset of a module that is also a module itself. It is closed under the module operations of addition and scalar multiplication.

## 2. What does it mean for a submodule to be finitely generated?

A submodule is finitely generated if it can be generated by a finite number of elements, meaning that every element in the submodule can be written as a linear combination of these generators.

## 3. What is a direct sum of left R-modules?

A direct sum of left R-modules is a construction where multiple modules are combined in a way that preserves their individual structures. This allows for the study of larger and more complex modules.

## 4. How do we know if a submodule is a direct sum of left R-modules?

A submodule is a direct sum of left R-modules if it can be written as the direct sum of two or more submodules, and each submodule has no nontrivial intersection with the others.

## 5. Why is the concept of a direct sum of left R-modules important?

The concept of a direct sum of left R-modules allows for the study of larger and more complex modules by breaking them down into smaller, more manageable submodules. It also provides a way to classify and understand the structure of modules in a systematic manner.

Replies
7
Views
2K
Replies
5
Views
1K
Replies
2
Views
1K
Replies
2
Views
1K
Replies
9
Views
2K
Replies
2
Views
2K
Replies
1
Views
948
Replies
1
Views
980
Replies
2
Views
2K
Replies
1
Views
2K