## Rings, Maximal and Principal Ideals

### Maximal and Principal Ideals

Like subgroups, an ideal H is maximal if
no ideal properly contains H and remains a proper subset of the ring.
A largest ideal is maximal, and contains all other ideals.

Since 0 is always an ideal, a minimal ideal is understood to be nonzero.
Of course there are maximal and minimal left/right ideals.

The union of an ascending chain of proper ideals remains a proper ideal,
since none of the ideals contains 1.
Use zorn's lemma
to extend a proper ideal H up to a maximal ideal that contains H.
Of course this can be done for left or right ideals.

A principal ideal is generated by a single element x in the ring.
If R is commutative, the principal ideal becomes x*R.
A noncommutative ring can have principal left ideals R*x and principal right ideals x*R.
A principal two-sided ideal contains all the elements R*x*R,
and all finite sums thereof.

In an integral domain,
the ideal generated by x is maximal in the collection of principal ideals
iff x is irreducible.
(This doesn't mean its a maximal ideal; it is maximal in the collection of principal ideals.)
If x is irreducible and the ideal generated by y contains the ideal generated by x,
y divides x,
and y or x/y is a unit.
The new ideal is the old ideal or the entire ring.
Conversely, if x generates a maximal principal ideal,
and x = a*b,
and a and b are nonunits,
then a generates a larger, proper, principal ideal
that contains x.
But how do we know it's larger?
If xc = a, then xcb = x, 1-cb is a zero divisor, and b becomes a unit.
That's why we need an integral domain, not just a commutative ring.