Results 1 to 4 of 4

Math Help - Unique R-Module Homomorphism

  1. #1
    Super Member Bernhard's Avatar
    Joined
    Jan 2010
    From
    Hobart, Tasmania, Australia
    Posts
    559
    Thanks
    2

    Unique R-Module Homomorphism

    I am reading Dummit and Foote Ch 10: Modules and in particulat section 10.3 Generation of Modules, Direct Sums and Free Modules.

    I am having problems understanding and interpreting Theorm 6 - see attached pdf for the two relevant pages of D&F.

    Problems I have are as follows:

    1. We are told to "identify A as a subset of F(A) by a  \mapsto  f_a where  f_a is the function which is 1 at a and zero elsehwhere"

    Thus  f_a  \in F(A) - but how can A be a subset of F(A)? The members of F(A) are functions  f_a ,  f_b ,  f_c , ... etc. The entries are functions not just simple points or elements of A. The nature of the elements of A and F(A) seem different, so can we "identify" the elements of A as a subset of F(A)?

    Further to this point where is the function a  \mapsto  f_a used as the proof progresses from this point?

    2. Following the quote above regarding  f_a , we read:

    "We can, in this way, think of F(A) as all finite R-linear combinations of elements of A by identifying each function f with the sum  r_1 a_1 +  r_2 a_2 + ..... +  r_n a_n where f takes the value  r_i at  a_i and is zero at all other elements of A. Moreover, each element of F(A) has a unique expression as such a formal sum."

    So you can see what assumptions I am working on, I have sketched my understanding of functions f, g  \in F(A) and also my understanding of  f_a - see pdf attached called sketches of functions

    Given that D&F describe f as taking on the value  r_i at  a_i and being zero at all other elements, how can we show that each element of F(A) has a unique expression as a formal sum of the form  r_1 a_1 +  r_2 a_2 + ..... +  r_n a_n .

    [I am also somewhat confused by the fact that the functional values of f are  r_1 ,  r_2 , etc and not  r_1 a_1 ,  r_2 a_2 , etc ]

    Can anyone help clarify this theorem for me?

    Peter
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7

    Re: Unique R-Module Homomorphism

    Quote Originally Posted by Bernhard View Post
    I am reading Dummit and Foote Ch 10: Modules and in particulat section 10.3 Generation of Modules, Direct Sums and Free Modules.

    I am having problems understanding and interpreting Theorm 6 - see attached pdf for the two relevant pages of D&F.

    Problems I have are as follows:

    1. We are told to "identify A as a subset of F(A) by a  \mapsto  f_a where  f_a is the function which is 1 at a and zero elsehwhere"

    Thus  f_a  \in F(A) - but how can A be a subset of F(A)? The members of F(A) are functions  f_a ,  f_b ,  f_c , ... etc. The entries are functions not just simple points or elements of A. The nature of the elements of A and F(A) seem different, so can we "identify" the elements of A as a subset of F(A)?

    Further to this point where is the function a  \mapsto  f_a used as the proof progresses from this point?

    2. Following the quote above regarding  f_a , we read:

    "We can, in this way, think of F(A) as all finite R-linear combinations of elements of A by identifying each function f with the sum  r_1 a_1 +  r_2 a_2 + ..... +  r_n a_n where f takes the value  r_i at  a_i and is zero at all other elements of A. Moreover, each element of F(A) has a unique expression as such a formal sum."

    So you can see what assumptions I am working on, I have sketched my understanding of functions f, g  \in F(A) and also my understanding of  f_a - see pdf attached called sketches of functions

    Given that D&F describe f as taking on the value  r_i at  a_i and being zero at all other elements, how can we show that each element of F(A) has a unique expression as a formal sum of the form  r_1 a_1 +  r_2 a_2 + ..... +  r_n a_n .

    [I am also somewhat confused by the fact that the functional values of f are  r_1 ,  r_2 , etc and not  r_1 a_1 ,  r_2 a_2 , etc ]

    Can anyone help clarify this theorem for me?

    Peter
    "identifying" here means that the map \psi:A \longrightarrow F(A) defined by \psi(a)=f_a is one-to-one. this is clear because if f_a=f_b and a \neq b, then 1=f_a(a)=f_b(a)=0, which is non-sense.

    so, using the above identification, r_1a_1 + \ldots + r_na_n will mean r_1f_{a_1} + \ldots r_nf_{a_n}.

    now let f \in F(A). then, by definition, f(a)=0 for all but finitely many a \in A. so suppose that B=\{a_1, \ldots , a_n\} \subseteq A is the set of all elements at which f is non-zero. let f(a_i)=r_i, \ 1 \leq i \leq n. let g = \sum_{i=1}^n r_i f_{a_i}. we claim that f = g. to see this consider two cases:
    case 1. x \in B: in this case, x = a_j, for some 1 \leq j \leq n, and thus

    f(x)=f(a_j)=r_j=r_jf_{a_j}(a_j)= \sum_{i=1}^n r_if_{a_i}(a_j)=g(a_j)=g(x).

    case 2. x \notin B : in this case f(x)=0=\sum_{i=1}^n r_if_{a_i}(x)=g(x).

    to see that why the representation is unique, suppose that f = \sum_{i=1}^n r_if_{a_i}=\sum_{i=1}^n s_i f_{a_i}. then, for any 1 \leq j \leq n:

    r_j=f(a_j)=\sum_{i=1}^nr_if_{a_i}(a_j)=\sum_{i=1}^  n s_if_{a_i}(a_j)=s_j.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member Bernhard's Avatar
    Joined
    Jan 2010
    From
    Hobart, Tasmania, Australia
    Posts
    559
    Thanks
    2

    Re: Unique R-Module Homomorphism

    Thanks NonCommAlg

    I would have never got that working by myself!

    Peter
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member Bernhard's Avatar
    Joined
    Jan 2010
    From
    Hobart, Tasmania, Australia
    Posts
    559
    Thanks
    2

    Re: Unique R-Module Homomorphism

    Thanks again ... but your clarification still left me wondering ...

    Your clarification seems to indicate that for x  \in A that

    f(x) = f(  a_j ) =  r_j

    and so

    g(x) =  \Sigma  r_i f_a_i (x) =  r_j

    Does this mean all sums  r_1 a_1 +  r_2 a_2 + ..... +  r_n a_n collapse to  r_j for some index j

    How do we get some 'genuine' sums  r_1 f_a_1 +  r_2 f_a_2 + ..... +  r_n f_a_n where  f_a_i is not equal to zero for many i or indeed for all i between 1 and n.

    so that  r_1 f_a_1 +  r_2 f_a_2 + ..... +  r_n f_a_n can be identified with a sum  r_1 a_1 +  r_2 a_2 + ..... +  r_n a_n where  a_i is not equal to zero for many or all i

    Peter
    Last edited by Bernhard; September 2nd 2011 at 10:30 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Z-module homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: November 28th 2009, 04:55 AM
  2. simple question on R module homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 23rd 2009, 08:12 AM
  3. projective, module homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 13th 2009, 06:57 PM
  4. module homomorphism, ring, subset
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 18th 2008, 11:00 PM
  5. R-module homomorphism surjectivity
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 9th 2008, 01:32 AM

Search Tags


/mathhelpforum @mathhelpforum