Ideal (ring Theory) - Ideal Operations

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.