Ideal Operations
The sum and product of ideals are defined as follows. For and, ideals of a ring R,
and
i.e. the product of two ideals and is defined to be the ideal generated by all products of the form ab with a in and b in . The product is contained in the intersection of and .
The sum and the intersection of ideals is again an ideal; with these two operations as join and meet, the set of all ideals of a given ring forms a complete modular lattice. Also, the union of two ideals is a subset of the sum of those two ideals, because for any element a inside an ideal, we can write it as a+0, or 0+a, therefore, it is contained in the sum as well. However, the union of two ideals is not necessarily an ideal.
Read more about this topic: Ideal (ring Theory)
Famous quotes containing the words ideal and/or operations:
“The ideal college is Mark Hopkins on one end of a log and a student on the other.”
—James A. Garfield (1831–1881)
“It may seem strange that any road through such a wilderness should be passable, even in winter, when the snow is three or four feet deep, but at that season, wherever lumbering operations are actively carried on, teams are continually passing on the single track, and it becomes as smooth almost as a railway.”
—Henry David Thoreau (1817–1862)