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Math Help - Submodules generated by A

  1. #1
    Super Member Bernhard's Avatar
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    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
    Last edited by Bernhard; August 10th 2013 at 07:47 PM.
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  2. #2
    Super Member Bernhard's Avatar
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    Re: Submodules generated by A

    Quote Originally Posted by Bernhard View Post
    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  A \subseteq N

    We need to show that  RA \subseteq N

    Let  x \in RA

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

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

    and then the addition of these elements, visually,  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  x \in N

    Thus  x \in RA \Longrightarrow x \in N

    So  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
    Last edited by Bernhard; August 10th 2013 at 08:28 PM.
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