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