Re: Ideals and Containment.

If L = M, then we're done. So, assume L properly contains M. That means there's an element x that's in L but not in M.

From the given, what can you say about x?

What can you say about about an ideal that contains a unit?

Re: Ideals and Containment.

Re: Ideals and Containment.

From the given, r is in M or r is a unit, but not both, right?

So, from where I left off, that means x is a unit, since it's not in M.

After, that yes, your reasoning is sound :) ... except, I think should be (with some re-wording).

Basic idea:

x is a unit so it has a multiplicative inverse (denoted 1/x, which is in R).

Since x is in L and L is an ideal, then xr is in L for all r in R.

That means x*(1/x) = 1 is in L (since 1/x is in R).

So, 1r = r is in L for all r in R.

Which gives L = R.