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: ,

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This entry was posted on 4 April, 2008 at 7:47 and is filed under undergrad maths. You can follow any responses to this entry through the RSS 2.0 feed.
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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 |

[…] Here I wrote more about various other operations on ideals. […]