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    Ideal rings

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    Note: Z is integer numbers
    C is set containment

    Here is the problem

    Let I be an ideal in a ring R.
    Define [ R : I ] = { r in R such that xr in R for all x in R }
    1) Show that [ R : I ] is an ideal of R that contains I
    2) If R is assumed to have a unity, what can you say about [ R : I ] ?
    3) Find [ 2Z : 12Z ], where 2Z is the ring of even integers

    For part 1) Please, pay careful attention to all the property you must verify to show that something is an ideal in a ring that is not assumed to be commutative.
    For example,
    i) closure under addition
    ii) 0 in [ R : I ]
    iii) If r in [ R : I ], then -r (inverse of r) in [ R : I ]
    iv) aN C N, and Nb C N where a, b in R and N is additive subgroup of a ring R

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    Solution Preview

    Definitons, from Fraleigh "A First Course in Abstract Algebra" (a highly recommended reference):

    Using C= to mean "subset of or equal to"

    A subring N of a ring R satisfying the properties
    aN C= N and Nb C= N for all a,b elements of R is an *ideal*

    A *ring* <R, +, *> is a set R together with two binary operations + and * which we call addition and multiplication, defined on R such that the following axioms are satisfied:
    <R,+> is an abelian group
    Multiplication is associative
    For all a, b, c elements of R, the left and right distributive laws,
    a(b + c) = ab + ac, (a + b) c = ac + bc, hold

    Note that Fraleigh does *not* require a ring to have a multiplicative identity 1 -- there are structures without a 1, for example 2Z ( but R must have a zero as an abelian additive group)

    He doesn't say specifically that the ring is closed under multiplication, but it must be (and he does state this in his next example) or we would get into terrible tangles.

    (1) You typed:
    Let I be an ideal in a ring R.
    Define [ R : I ] = { r in R such that xr in R for all x in R }

    Must be a typo. This isn't going to work -- there has to be an I in that definition somewhere. As stated, it's a ...

    Solution Summary

    This is a proof regarding ideal rings. The set of containments are determined.