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Commutative ring theorem

  1. Dec 1, 2009 #1
    1. The problem statement, all variables and given/known data
    I need to prove this theorem Let R be a commutative ring with identity and c1,c2,...cn E (element of) R Then the set I={r1c1+r2c2+....+rncn|r1,r2,...,rn E R} is an ideal in R




    2. Relevant equations

    Well I do know I need to prove closure under subtraction, closure under multiplication and absorption r E R and a E I show ra E I and ar E I I know because R is commutative I do not have to prove both ways. and I know that just proving absorption will prove multiplication.

    3. The attempt at a solution

    so, I guess I am just confused on what the set I looks like.

    to prove closure under subraction I need to elements from I. What would these two elements look like?
     
  2. jcsd
  3. Dec 2, 2009 #2

    HallsofIvy

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    The set "looks like" exactly what it says: All things of the form [itex]r_1c_1+ r_2c_2+ \cdot\cdot\cdot+ r_nc_n[/itex] where [itex]c_1, c_2, \cdot\cdot\cdot, c_n[/itex] are given (fixed) members of ring R and [itex]r_1, r_2, \cdot\cdot\cdot, r_n[/itex] can be any members of the ring.

    So you want to prove:
    (1) Closed under addition: If [itex]r_1c_1+ r_2c_2+ \cdot\cdot\cdot+ r_nc_n[/itex] and [itex]s_1c_1+ s_2c_2+ \cdot\cdot\cdot+ s_nc_n[/itex] are members of this set, it their sum also a member? That is can it be written in the same way: find [itex]t_1, t_2, \cdot\cdot\cdot, t_n[/itex] such that the sum of those two is [itex]t_1c_1+ t_2c_2+ \cdot\cdot\cdot+ t_nc_n[/itex].

    (2) "Absorption" of the entire ring: If [itex]r_1c_1+ r_2c_2+ \cdot\cdot\cdot+ r_nc_n[/itex] is in the set, show that [itex]r(r_1c_1+ r_2c_2+ \cdot\cdot\cdot+ r_nc_n)[/itex], for r any member of the ring, is of that same form.

    Those are both just as easy as they look.
     
  4. Dec 2, 2009 #3
    Ok that is easy.

    So under addition, because R is a ring then r1+s1 will be in R so the (r1+s1)c1+(r2+s2)c2... is in I

    And because R is a Ring and closed under multiplication r*r1, r*r2,.... are in R so r*r1c1+r*r2c2+..... is in I

    So it is a ideal in R

    Thanks
     
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