Assignment:
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
Provide complete and step by step solution for the question and show calculations and use formulas.