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:
“If we love-and-serve an ideal we reach backward in time to its inception and forward to its consummation. To grow is sometimes to hurt; but who would return to smallness?”
—Sarah Patton Boyle, U.S. civil rights activist and author. The Desegregated Heart, part 3, ch. 3 (1962)
“You can’t have operations without screams. Pain and the knife—they’re inseparable.”
—Jean Scott Rogers. Robert Day. Mr. Blount (Frank Pettingell)