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.