Guiding Questions.
In this section, we’ll seek to answer the questions:
- What are ideals, and what can we do with them?
Section 1.5 Ideals
One of the ways in which mathematicians study the structure of an abstract object is by considering how it interacts with other (related) objects. This is especially true of its subobjects. Thus, in linear algebra, we are often concerned with subspaces of a vector space as a means of understanding the vector space, or even submatrices as a way of understanding a matrix (see, e.g., the cofactor expansion formula for the determinant). In real analysis and topology, the important subobjects are usually open sets, or subsequences, and the study of a graph’s subgraphs is an important approach to many questions in graph theory. In algebra, we learn about a ring by studying its relationship to other rings via functions (introduced in Section 3.1) and to its ideals, introduced below in Definition 1.5.1.
The notion of an ideal number was first introduced by Ernst Kummer in the middle of the nineteenth century. Kummer was studying the cyclotomic integers in connection to work on Fermat’s Last Theorem and reciprocity laws in number theory, and discovered, to use our modern terminology, that these rings of cyclotomic integers were not UFDs. In particular, he found irreducible cyclotomic integers that were not prime. His work, which was finished by Richard Dedekind by 1871, was to define a new class of complex number, an ideal number for which unique factorization into prime ideal numbers held. A related notion was developed by Kronecker and Lasker, before the two concepts were unified by David Hilbert and Emmy Noether into the more general version which we explore here. We then use ideals to build new classes of rings, known as quotient rings.
Definition 1.5.1.
A subset \(I\) of a (not necessarily commutative) ring \(R\) is called an ideal if:
- \(\displaystyle 0\in I\)
- for all \(x,y\in I\text{,}\) \(x+y\in I\text{;}\) and,
- for all \(x\in I\) and for all \(r\in R\text{,}\) \(xr\in I\) and \(rx\in I\text{.}\)
Observe that the third requirement for a set \(I\) to be an ideal of \(R\) is simplified slightly if \(R\) is commutative (which, we recall, all of our rings are).
There are many important examples and types of ideals, but there are also some trivial ideals contained in every ring.
Theorem 1.5.2.
Let \(R\) be a ring. Then \(R\) and \(\set{0}\) are ideals of \(R\text{.}\)
Theorem 1.5.3.
All ideals are subrings.
The following theorem provides a useful characterization of when an ideal \(I\) is in fact the whole ring.
Theorem 1.5.4.
Let \(R\) be a ring and \(I\) an ideal of \(R\text{.}\) Then \(I = R\) if and only if \(I\) contains a unit of \(R\text{.}\)
Investigation 1.5.5.
Consider the ring \(R = \Z\text{,}\) and the sets \(I = \setof{2x}{x\in \Z}\) and \(J = \setof{3x}{x\in\Z}\text{.}\)
- Prove that \(I\) and \(J\) are ideals of \(R\text{.}\)
- Describe the set \(I\cap J\text{.}\) Is \(I\cap J\) an ideal of \(R\text{?}\)
- Describe the set \(I\cup J\text{.}\) Is \(I\cup J\) an ideal of \(R\text{?}\)
We next explore the behavior of ideals under the usual set-theoretic operations of intersection and union.
Theorem 1.5.6.
Let \(R\) be a ring and let \(\set{I_{\alpha}}_{\alpha\in \Gamma}\) be a family of ideals. Then \(I = \bigcap\limits_{\alpha\in \Gamma} I_\alpha\) is an ideal.
Investigation 1.5.7.
Let \(R\) be a ring and \(I,J\subseteq R\) be ideals. Must \(I\cup J\) be an ideal of \(R\text{?}\) Give a proof or counterexample of your assertion.
In addition to the set-theoretic properties described above, we can do arithmetic with ideals.
Theorem 1.5.8.
Let \(R\) be a ring and \(I,J\subseteq R\) ideals of \(R\text{.}\) Then the sum of \(I\) and \(J\text{,}\)
\begin{equation*}
I+J := \setof{x+y}{x\in I, y\in J},
\end{equation*}
is an ideal of \(R\text{.}\) Furthermore, the product of \(I\) and \(J\text{,}\)
\begin{equation*}
IJ := \setof{x_1 y_1 + x_2 y_2 + \cdots + x_n y_n}{n\ge 1, x_i \in I, y_j\in J}
\end{equation*}
is an ideal of \(R\text{.}\)
The most important type of ideals (for our work, at least), are those which are the sets of all multiples of a single element in the ring. Such ideals are called principal ideals.
Theorem 1.5.9.
Let \(R\) be commutative with identity and let \(a\in R\text{.}\) The set
\begin{equation*}
\ideal{a} = \setof{ra}{r\in R}
\end{equation*}
is an ideal (called the principal ideal generated by \(a\)).
The element \(a\) in the theorem is known as a generator of \(\ideal{a}\text{.}\)
Activity 1.5.10.
In \(R = \Z\text{,}\) describe the principal ideals generated by
- 2
- \(\displaystyle -9\)
- 9
- 0
- 27
- 3
Determine the subset relations among the above ideals.
Investigation 1.5.11.
Let \(R\) be commutative with identity, and let \(x,y,z\in R\text{.}\) Give necessary and sufficient conditions for \(z\in \ideal{x}\) and, separately, \(\ideal{x} \subseteq \ideal{y}\text{.}\)
That is, fill in the blanks: “\(z\in \ideal{x} \Leftrightarrow\) _________” and “\(\ideal{x}\subseteq \ideal{y} \Leftrightarrow\) _________.”
Justify your answers.
Principal ideals may have more than one generator.
Theorem 1.5.12.
Let \(R\) be a ring and \(a\in R\text{.}\) Then \(\ideal{a} = \ideal{ua}\text{,}\) where \(u\) is any unit of \(R\text{.}\)
While principal ideals have a satisfyingly simple structure, not every ideal is principal (see the Challenge 1.8.4). Still, we would like a way to more precisely describe the elements of such ideals; we begin with Definition 1.5.13.
Definition 1.5.13.
Let \(R\) be a commutative ring with identity, and let \(S\subseteq R\) be a subset. Then
\begin{equation}
\langle S \rangle := \bigcap\limits_{\substack{J\supseteq S\\\text{ \(J\) is an ideal } } } J\tag{1.5.1}
\end{equation}
is called the ideal generated by \(S\) , and we call \(S\) the generating set for the ideal.
A consequence of Definition 1.5.13 is the following theorem.
Theorem 1.5.14.
Let \(R\) be a ring. Then \(\ideal{\emptyset} = \set{0}\text{.}\)One way to interpret Definition 1.5.13 is that \(\ideal{S}\) is the smallest ideal (with respect to subset inclusion) that contains \(S\text{.}\)
Theorem 1.5.15.
Given a commutative ring \(R\) and a subset \(S\) of \(R\text{,}\) \(\ideal{S}\) is the smallest ideal containing \(S\) in the sense that, if \(J\) is any ideal of \(R\) containing \(S\text{,}\) \(\ideal{S}\subseteq J\text{.}\)
The concept elucidated by Theorem 1.5.15 is helpful, but does not give us a handle on the structure of the elements of \(\ideal{S}\text{.}\) Such a description is provided by Theorem 1.5.16.
Theorem 1.5.16.
Given a commutative ring with identity \(R\) and a nonempty subset \(S\) of \(R\text{:}\)
- The set \(I = \setof{r_1 s_1 + r_2 s_2 + \cdots + r_n s_n}{r_i\in R, \ s_j \in S,\ n\ge 1}\) is an ideal of \(R\text{;}\)
- \(S\subseteq I\text{;}\) and
- \(I = \ideal{S}\text{.}\)
In other words, the ideal \(\ideal{S}\) contains all possible finite “\(R\)-linear combinations” of elements of \(S\text{;}\) that is, it contains all finite sums of products of ring elements with elements from \(S\text{.}\)
Definition 1.5.17.
If \(R\) is a ring and \(S = \set{s_1, s_2, \ldots,
s_n}\) is a finite subset of \(R\text{,}\) the ideal \(I\) generated by \(S\) is denoted by \(I = \ideal{s_1, s_2, \ldots,
s_n}\text{,}\) and we say \(I\) is finitely generated .
We close with a discussion of a class of ideals which are the object of active mathematical research. Recall that a (simple) graph \(G\) consists of a set \(V = \set{x_1, x_2, \ldots, x_n}\) of vertices together with a collection \(E\) of edges, which are just pairs of vertices and can be written \(x_i x_j\text{.}\) This notation suggests the following definition.
Definition 1.5.18.
Let \(K\) be a field, \(G\) a graph on the vertex set \(V = \set{x_1, x_2, \ldots, x_n}\) with edge set \(E\text{,}\) and let \(R = K[x_1, x_2, \ldots, x_n]\) be the ring of polynomials whose variables are the vertices of \(G\) with coefficients in \(K\text{.}\) Define the edge ideal of \(G\) to be
\begin{equation*}
I(G) := \ideal{x_i x_j \mid x_i x_j\in E}.
\end{equation*}
That is, \(I(G)\) is generated by the products of the variables corresponding to the edges of the graph.
Activity 1.5.19.
Consider the graph \(G\) in Figure 1.5.20. List the generators of \(I(G)\) and an appropriate ring in which \(I(G)\) may live.
As one might hope, we do not make Definition 1.5.18 merely for fun; given a graph \(G\text{,}\) it is possible to relate the graph-theoretic properties of \(G\) (e.g., the chromatic number) with the ideal-theoretic properties of \(I(G)\text{.}\) See [1.5.1] and [1.5.2], among others, for more.
References References
[1]
A. Van Tuyl, A Beginner’s Guide to Edge and Cover Ideals, in Monomial Ideals, Computations, and Applications, Lecture Notes in Mathematics Volume 2083, 2013, pp 63-94
2
dx.doi.org/doi:10.1007/978-3-642-38742-5_3[2]
C. Bocci, S. Cooper, E. Guardo, et al., The Waldschmidt constant for squarefree monomial ideals, J Algebr Comb (2016) 44:875
3
doi.org/10.1007/s10801-016-0693-7
