Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Alternate expression for definition of an Ideal?

  1. Feb 22, 2015 #1
    If A is an additive subgroup of a ring R, A is said to be an ideal if Ra is contained in A for all a in A; that is, if every multiple of an element of A is again in A.

    Is it true that A is an ideal of R if Ar is contained in A for all r in R? To me it seems like they are equivalent statements but I'm not sure.

    Update: Second thought, I believe these are NOT equivalent statements. I'm looking at the proof that explains why if 1 is in A then A = R, and you could not employ this argument under the faulty definition I tried to push. Anyone wanna shed further light on the difference between these statements?
     
  2. jcsd
  3. Feb 24, 2015 #2

    Stephen Tashi

    User Avatar
    Science Advisor

  4. Feb 24, 2015 #3
    Ah right, didn't mean to switch those around. Would it be fair to say that the following statements define the same ideal A?

    Ar is contained in A for all r in R
    aR is contained in A for all a in A
     
  5. Feb 24, 2015 #4

    lavinia

    User Avatar
    Science Advisor

    What makes you think that they are the same?
     
  6. Feb 24, 2015 #5

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Assuming that ##A## is an additive subgroup of ##R##, I think either of these conditions implies that ##A## is a right ideal.

    In either case, we need to verify that if ##a \in A## and ##r \in R##, then ##ar \in A##.

    Suppose the first condition holds. If ##a \in A## and ##r \in R##, then ##ar \in Ar \subseteq A##, so ##A## is a right ideal.

    Now suppose the second condition holds. If ##a \in A## and ##r \in R##, then ##ar \in aR \subseteq A##, so ##A## is a right ideal.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Alternate expression for definition of an Ideal?
  1. Alternating group (Replies: 2)

Loading...