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Math Help - Examples of Ideals

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    Examples of Ideals

    If i want to find out if {(n,n):n in Z} is an ideal of ZxZ, is there a test or prperties i can apply to prove it is an ideal? My book mentions something about finding a homomorphism? i dont know...
    Last edited by Aquameatwad; April 30th 2012 at 07:50 PM.
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    Re: Examples of Ideals

    two magic steps to proving J is an ideal of a ring R:

    let a be any element of R, x,y be any elements of J

    1) show x - y is in J
    2) show ax and xa are in J

    that's it!

    let's apply these to the set J = {(n,n): n in Z}.

    let a = (b,c), x = (k,k), y = (m,m).

    x - y = (k,k) - (m,m) = (k-m,k-m). check.

    ax = (b,c)(k,k) = (bk,ck)...uh oh, it appears we DON'T have an ideal (because b might not equal c, so bk might not equal ck).

    let's try "the homomorphism approach": J is an ideal of R, iff J = ker(φ) for some (ring) homomorphism φ:R→S

    if that were so, we must have φ((k,k)) = 0S, for every k in Z, and φ((k,m)) ≠ 0S, if k ≠ m

    (we want the kernel to be J, and ONLY J).

    now note that if b ≠ c, then bk ≠ ck, but (since φ is a ring homomorphism):

    φ((bk,ck)) = φ((b,c))(k,k)) = φ((b,c))φ((k,k)) = φ((b,c))0S = 0S.

    but (bk,ck) isn't in J, so any ring homomorphism that "kills" J, kills MORE than J: J is not an ideal.

    the way i think of ideals is this: ideals are characterized by a property that "absorbs" all of R.

    one example is "even-ness" in the integers. the even numbers form an additive subgroup of Z (this is #1 above, we use the "one-step subgroup test" on (J,+)). and no matter what integer we multiply an even integer by, the result is always even. "even-ness" is "contagious" when you multiply.

    the properties of ideals "generalize" the following properties of 0:

    0 - 0 = 0
    a0 = 0a = 0

    so an ideal can be thought of as: "making a bigger 0-type of thingy" (which is exactly what we DO when we form the quotient ring R/J, we let J play the part of "0". and if it's going to play that part, then goshdagnabbit, it'd better have the right properties!).

    the "typical" example is Z/nZ, where we let "multiples of n" play 0. so the integers get "reset" every n spaces, and go around in a circle, instead of a line. and yes, nZ is an ideal of Z (the ideal generated by n, in fact, so Z/nZ = Z/(n) = Zn. it's all very cozy).
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    Re: Examples of Ideals

    For magic step 2, why do we choose a=(b,c), but x=(k,k) and y=(m,m)? why isnt it just a=(b,b)? i guess, why are there 2 diff elements instead of just one
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    Re: Examples of Ideals

    because a has to be a "completely arbitrary" element of Z x Z (and not just one of the "special ones" of the form (k,k)). look at rule #2 of an ideal.
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    Re: Examples of Ideals

    would it be similar to solve if {(n,m); n+m is even} is an ideal?

    so we have a=(b,c), x=(k,r), y=(m,s)

    1.) x-y= (k,r)-(m,s)= (k-m, r-s)

    Now wouldn't this not be an ideal since k and r have to be either both even or both odd and m and s have to be either both even or both odd, so for example k-m and r-s have the chance of being even + odd which would not equal even
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    Re: Examples of Ideals

    you're on the right track, but you made a wrong turn somewhere. if J = {(n,m) in Z x Z: n+m = 2u, for some u in Z}, and if x,y are in J, then:

    x = (k,r) where k+r = 2u
    y = (m,s) where m+s = 2v.

    therefore: x-y = (k,r) - (m,s) = (k-m,r-s), and k-m + r-s = (k+r) - (m+s) = 2u - 2v = 2(u-v), hence x-y is in J.

    so condition #1 IS satisfied.

    let's look at condition #2. let a = (b,c) (where b and c might be any integers).

    then ax = (b,c)(k,r) = (bk,cr). can we say that bk + cr is even? well, no.

    suppose (k,r) = (1,3), which is clearly in J, and suppose a = (2,1). then (2,1)(1,3) = (2,3), and 2+3 = 5 is odd. so we don't have an ideal, but not for the reason you gave.

    it's usually condition #2 that is the "tricky" part.
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    Re: Examples of Ideals

    with all these theorems and lemmas an such i forget how to do basic math...
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