Given a commutative ring R, one can do different operations with ideals. If you are familiar with operations on subgroups in groups, a number of these thing will look very familiar.
The easiest operation is ideal intersection, that we have already encountered: given two ideals , their intersection is an ideal itself. Indeed, the intersection of subgroups (we talk about additive subgroups of the additive group of R) is a subgroup, and for any and one has
and , implying , as required from to be an ideal.
Similarly, we can see that the intersection of an arbitrary number of ideals is an ideal.
Note that the union of two ideals need not be an ideal: e.g. is not an ideal, as it is already not a subgroup of the additive group of R, as can be seen by taking , and computing .
The rest of the operations are of more algebraic flavour. The first is the ideal sum of two ideals . By definition, . This is indeed an ideal: it is a subgroup of the additive group of R, and also for any . As well, note that is the minimal w.r.t. inclusion ideal of R that contains .
One can generalise this to the sum
of an arbitrary collection of ideals: take finite
(in sense that they have only finitely many nonzero summands) sums for
the set of elements of . Again, we note that is the
minimal ideal containing .
The second operation is ideal product of two ideals . By definition, , that is, the ideal generated by the products , where . This is an ideal, just by definition. Note that .
Moreover, iff and are coprime, i.e.
Indeed, It is easy to check that
this product is associative, and we can drop brackets:
A particular important kind of ideal product is the -th power of an ideal
By definition , where the product is taken times.
The third operation is in some sense the inverse of multipication, it is ideal quotient
. In particular is called the annihilator of
You are invited to check the following properties of the ideal quotient: ,
13 April, 2008 at 19:37 |
cool!
i’ve added the page to my bookmarks. looks like it’s gonna be interesting..=)
19 February, 2009 at 13:16 |
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