A subring * I * of a ring
is called a left ideal if given any element * x * of the subring
* I * the product * yx * is in * I * for any choice
of * y * from the original ring. An analogous definition is made
for right ideals with respect to multiplication on the right. If an ideal
is both a left and right ideal it is called a two-sided ideal or simply
an ideal. The kernel of a (ring) homomorphism provides the canonical
example of an ideal.

A technique for counting the elements of a set that *don't* have
certain properties. Let * S * be a set of *n*-elements,
and let * P1 * and * P2 * be properties that the elements
of * S * may or may not have. Let * A1 * be the subset of
* S * consisting of those elements having * P1 * and
* A2 * be those having * P2 *.
The number of elements of * S * having neither * P1 * nor
* P2 * is given by:

* |S|-|A1|-|A2|+|A1 *intersect * A2|*

The last term has to be added because the elements having both properties have been subtracted twice in the previous terms.

This is the general idea of the inclusion/exclusion
principle, the formula becomes slightly more complicated but can readily
be generalized to *n* properties.

In a directed graph, we say that a vertex has indegree *x* if
there are (exactly) *x* edges coming into that vertex.

A set of vertices in a graph none of which are connected by an edge.

An element of order 2 in a group.

A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z

An On-line Dictionary of Combinatorics