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- Jun 22, 2012

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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 ... "

I am not sure how to

However, reflecting on the above, it is easy to show that RA is a submodule of M, but what is worrying me is the formal proof that RA is the

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 [TEX] A \subseteq N [/TEX]

We need to show that [TEX] RA \subseteq N [/TEX]

Let [TEX] x \in RA [/TEX]

Now [TEX]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}^{+} [/TEX]

So [TEX] x = r_1a_1 + r_2a_2 + ... ... + r_ma_m [/TEX] for [TEX] r_1, r_2, ... ... , r_m \in R, \ a_1, a_2, ... ... , a_m \in A [/TEX]

If [TEX] A \subseteq N [/TEX] then [TEX] r_ia_i \in N [/TEX] for [TEX] 1 \le i \le n [/TEX] since N is a submodule

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

So [TEX] x \in N [/TEX]

Thus [TEX] x \in RA \Longrightarrow x \in N [/TEX]

So [TEX] A \subseteq N \Longrightarrow RA \subseteq N [/TEX] ... ... (1)

However, I am still not completely sure how to formally show that RA is

Peter

[This has also been posted on MHF]

"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 ... "

I am not sure how to

__prove this statement.__**(formally and explicitly)**However, reflecting on the above, it is easy to show that RA is a submodule of M, but what is worrying me is the formal proof that RA is the

**submodule of M that contains A.***smallest*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 [TEX] A \subseteq N [/TEX]

We need to show that [TEX] RA \subseteq N [/TEX]

Let [TEX] x \in RA [/TEX]

Now [TEX]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}^{+} [/TEX]

So [TEX] x = r_1a_1 + r_2a_2 + ... ... + r_ma_m [/TEX] for [TEX] r_1, r_2, ... ... , r_m \in R, \ a_1, a_2, ... ... , a_m \in A [/TEX]

If [TEX] A \subseteq N [/TEX] then [TEX] r_ia_i \in N [/TEX] for [TEX] 1 \le i \le n [/TEX] since N is a submodule

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

So [TEX] x \in N [/TEX]

Thus [TEX] x \in RA \Longrightarrow x \in N [/TEX]

So [TEX] A \subseteq N \Longrightarrow RA \subseteq N [/TEX] ... ... (1)

However, I am still not completely sure how to formally show that RA is

__submodule of M that contains A i.e. how does the implication (1) demonstrate this - can someone help by showing this explicitly and formally?__**the smallest**Peter

[This has also been posted on MHF]

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