Submodules generated by A

On page 351 Dummit and Foote make the following statement:

"It is easy to see using the Submodules' Criterion that for any subset A of M, RA is indeed a submodule of M and is the smallest submodule of M which contains A ... "

(R is a ring with 1, M is a left module over R, A is a subset of M)

Can anyone formally and explicitly prove this statement?

Peter

Re: Submodules generated by A

Quote:

Originally Posted by

**Bernhard** On page 351 Dummit and Foote make the following statement:

"It is easy to see using the Submodules' Criterion that for any subset A of M, RA is indeed a submodule of M and is the smallest submodule of M which contains A ... "

(R is a ring with 1, M is a left module over R, A is a subset of M)

Can anyone formally and explicitly prove this statement?

Peter

Just reflecting on my own question - it is easy to show that RA is a submodule of M, but what was worrying me was the formal proof that RA is the *smallest* submodule of M that contains A.

However following the statement:

"It is easy to see using the Submodules' Criterion that for any subset A of M, RA is indeed a submodule of M and is the smallest submodule of M which contains A ... " Dummit and Foote write:

"i.e. any submodule of M which contains A also contains RA"

[I still find it perplexing that this actually shows that RA is the smallest submodule of M which contains A but anyway ... this is, I think, not hard to prove ...}

So to show that any submodule of M which contains A also contains RA

Let N be a submodule of M such that $\displaystyle A \subseteq N $

We need to show that $\displaystyle RA \subseteq N $

Let $\displaystyle x \in RA $

Now $\displaystyle RA = \{ r_1a_1 + r_2a_2 + ... ... + r_ma_m \ | \ r_1, r_2, ... ... , r_m \in R, \ a_1, a_2, ... ... , a_m \in A, m \in \mathbb{Z}^{+} $

So $\displaystyle x = r_1a_1 + r_2a_2 + ... ... + r_ma_m $ for $\displaystyle r_1, r_2, ... ... , r_m \in R, \ a_1, a_2, ... ... , a_m \in A $

If $\displaystyle A \subseteq N $ then $\displaystyle r_ia_i \in N $ for $\displaystyle 1 \le i \le n $ since N is a submodule

and then the addition of these elements, visually, $\displaystyle r_1a_1 + r_2a_2 + ... ... + r_ma_m $ also is in N (if two elements belong to a submodule then so does the element that is formed by their addition)

So $\displaystyle x \in N $

Thus $\displaystyle x \in RA \Longrightarrow x \in N $

So $\displaystyle A \subseteq N \Longrightarrow RA \subseteq N $ ... ... (1)

However, I am still not completely sure how to formally show that RA is the smallest submodule of M that contains A i.e. how does the implication (1) demonstrate this - can someone help by showing this explicitly and formally?

Peter