Archive for April, 2008

operations on ideals

4 April, 2008

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 I,J\subseteq R, their intersection I\cap J 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 x\in I\cap J and r\in R one has
rx\in I and rx\in J, implying rx\in I\cap J, as required from I\cap J 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. (2)\cup (3)\subseteq\mathbb{Z} is not an ideal, as it is already not a subgroup of the additive group of R, as can be seen by taking 3 \in (3), -2 \in (2) and computing 3+(-2)=1\not\in (2)\cup (3).

The rest of the operations are of more algebraic flavour. The first is the ideal sum I+J of two ideals I,J\subseteq R. By definition, I+J=\{x+y\mid x\in I, y\in J\}. This is indeed an ideal: it is a subgroup of the additive group of R, and also r(x+y)=rx+ry\in I+J for any r\in R. As well, note that I+J is the minimal w.r.t. inclusion ideal of R that contains I\cup J.
One can generalise this to the sum \sum_{\alpha\in A}I_\alpha
of an arbitrary collection A of ideals: take finite
(in sense that they have only finitely many nonzero summands) sums \sum_{x\alpha\in I_\alpha x_\alpha} for
the set of elements of \sum_{\alpha\in A}I_\alpha. Again, we note that \sum_{\alpha\in A}I_\alpha is the
minimal ideal containing \cup_{\alpha\in A}I_\alpha.

The second operation is ideal product I\cdot J of two ideals I,J\subseteq R. By definition, IJ=(xy\mid x\in I, y\in J), that is, the ideal generated by the products xy, where x\in I, y\in J. This is an ideal, just by definition. Note that IJ\subseteq I\cap J.
Moreover, IJ=I\cap J iff I and J are coprime, i.e.
I+J=(1). Indeed, (I+J)(I\cap J)=I(I\cap J)+J(I\cap J)\subseteq IJ. It is easy to check that
this product is associative, and we can drop brackets:
I\cdot(J\cdot K)=(I\cdot J)\cdot K=I\cdot J\cdot K.
A particular important kind of ideal product is the k-th power I^k of an ideal I.
By definition I^k=I\cdot I\cdot I\cdot\dots\cdot I, where the product is taken k times.

The third operation is in some sense the inverse of multipication, it is ideal quotient
(I:J)=\{x\in R\mid xJ\subseteq I\}. In particular (0:J) is called the annihilator of J.
You are invited to check the following properties of the ideal quotient: (I:I)=R, (I:R)=I.