Ideal and Factor Ring Problem: Proving A=R When 1 is an Element of A

[Benzoate]

Homework Statement

If A is an ideal of a ring R and 1 belongs to A, prove that A=R.

Homework Equations

The Attempt at a Solution

I said that r should an element of R. and since A is ideal to ring R and 1 is an element of A , then ar should be an element of A . 1 must be an element of a which is an element of ar which is an element of A. Therefore 1*ra=ar*1=> 1 is an element of R. Therefore,R=A

 

[Office_Shredder]

1 must be an element of a doesn't mean anything... what the heck is a supposed to be anyway? I'm assuming it's an element of A maybe... at any rate, nothing can be an element of ar as ar is simply a member of the ring, and you have no reason to believe it's a set.

You realize an ideal is defined such that if a is in A, then for all x in R, x*a is in A?