Section 1.6 Quotient Rings
Guiding Questions.
In this section, we’ll seek to answer the questions:
If the only rings that existed were polynomial rings, familiar systems of numbers like \(\Z, \Q, \R, \C\text{,}\) and matrix rings, there would still be enough to justify defining the concept of a ring and exploring its properties. However, these are not the only rings that exist. In this section, we explore a way of building new rings from old by means of ideals. To better understand these new rings, we will also define two new classes of ideals: prime ideals, and maximal ideals.
The major concept of this section is the notion of congruence modulo \(I\text{.}\) One can reasonably think of this idea as a generalization of congruence modulo \(m\) in \(\Z\text{.}\)
Definition 1.6.1.
Let \(R\) be a ring, \(I\) an ideal of \(R\text{,}\) and \(a,b\in R\text{.}\) We say \(a\) is congruent to \(b\) modulo \(I\) if \(b-a\in I\text{.}\) If this is the case, we write \(a + I = b + I\text{.}\)
Activity 1.6.2.
Determine (with brief justification) whether \(a + I = b + I\) in the following rings \(R\text{.}\)
\(a = 9\text{,}\) \(b = 3\text{,}\) \(I = \ideal{6}\text{,}\) \(R = \Z\)
\(a = 10\text{,}\) \(b = 4\text{,}\) \(I = \ideal{7}\text{,}\) \(R = \Z\)
\(a = 9+9x^2\text{,}\) \(b = 3-3x^2\text{,}\) \(I = \ideal{6}\text{,}\) \(R = \Z[x]\)
\(a = x^2+x-2\text{,}\) \(b = x-1\text{,}\) \(I = \ideal{x+1}\text{,}\) \(R = \Q[x]\)
(Challenge.) \(a = x^3\text{,}\) \(b = x^2+2x\text{,}\) \(I = \ideal{y-x^2, y-x-2}\text{,}\) \(R = \Q[x,y]\)
You may use the following Macaulay2 code cell to check your work!
Exploration 1.6.3.
Given a ring \(R\text{,}\) ideal \(I\text{,}\) and \(a\in R\text{,}\) when is it the case that \(a + I = 0 + I = I\text{?}\)
Observe that if \(b-a \in I\text{,}\) then there is some \(x\in I\) such that \(b-a = x\text{,}\) and so \(b = a+x\text{.}\)
Activity 1.6.4 demonstrates the sense in which congruence modulo
\(I\) is a generalization of congruence modulo
\(m\) in
\(\Z\text{.}\)
Activity 1.6.4.
Let \(m\in\Z\) with \(m \gt 1\) and consider the ideal \(I = \ideal{m}\) in \(\Z\text{.}\) Prove that \(a + I = b + I\) if and only if \(a\equiv b\mod m\text{.}\)
As was the case in \(\Z_m\text{,}\) congruence modulo \(I\) is an equivalence relation.
Theorem 1.6.5.
Let \(R\) be a ring and \(I\) an ideal of \(R\text{.}\) Then congruence modulo \(I\) is an equivalence relation on \(R\text{.}\)
The set of equivalence classes under this relation is denoted \(R/I\text{.}\) What is more, this is not merely a set of equivalence classes. As the next two theorems demonstrate, this set possesses two algebraic operations that extend naturally from those of \(R\text{.}\)
Theorem 1.6.6.
Let \(R\) be a ring and \(I\) an ideal of \(R\text{.}\) If \(a,b,c,d\in R\) such that \(a+I = b+I\) and \(c+I = d+I\text{,}\) then \((a+c) + I = (b+d) + I\text{.}\)
Theorem 1.6.7.
Let \(R\) be a ring and \(I\) an ideal of \(R\text{.}\) If \(a,b,c,d\in R\) such that \(a+I = b+I\) and \(c+I = d+I\text{,}\) then \(ac + I = bd + I\text{.}\)
The previous two theorems together show that addition and multiplication on the set
\(R/I\) is well-defined. As these operations are built on the operations of
\(R\text{,}\) it will likely not surprise you to learn that the usual
axioms defining a ring also hold.
Theorem 1.6.8.
Let \(R\) be a commutative ring with identity \(1_R\text{,}\) \(I\) an ideal of \(R\text{,}\) and \(a,b\in R\text{.}\) The set of equivalence classes modulo \(I\text{,}\) denoted \(R/I\text{,}\) is a commutative ring (called the quotient ring of \(R\) by \(I\)) with identity \(1_R + I\) under the following operations:
\begin{align*}
(a+I) + (b+I) & := (a+b) + I\\
(a+I) \cdot (b+I) & := (ab) + I.
\end{align*}
Thus, given a ring \(R\) and ideal \(I\) of \(R\text{,}\) we may build the quotient ring \(R/I\text{.}\)
For each of the following activities, be prepared to give an informed justification of your thinking.
Activity 1.6.9.
Suppose \(R= \Z\) and \(I = \ideal{8}\text{.}\) Calculate \((3+I) + (7+I)\) and \((4+I)\cdot (6+I)\text{.}\) Check your work using the M2 cell below.
Activity 1.6.10.
Suppose \(R= \Z[x]\) and \(I = \ideal{x^2}\text{.}\) Describe the elements of \(R/I\text{.}\) Is \(R/I\) an integral domain? If so, give a proof. If not, give an example of a zero divisor pair in \(R/I\) (you may use the M2 cell below to confirm your answer).
Activity 1.6.11.
Let \(R = \Z_3[x]\) and \(J = \ideal{x^2}\text{.}\) List all elements of \(R/J\) (there should be 9 in total).
Activity 1.6.12.
Set \(R = \R[x]\) and \(I = \ideal{x^2-1}\text{.}\) Find a nonzero element \(f + I \in R/I\) such that \((f+I)(2x+2 + I) = I\text{.}\) Confirm your answer below.
In
Section 3.2, we will explore the question of when
\(R/I\) possesses some of the properties we’ve previously explored, e.g., when is
\(R/I\) a domain? A field? First, we conclude with two explorations. The first gives us a sense of what these rings can look like. The second connects quotient rings to
solution sets of polynomial equations.
Exploration 1.6.13.
Consider the ring \(R=\Z_2[x]\) and the ideals \(I = \ideal{x^2-1}\) and \(J = \ideal{x^3 -x -1}\text{.}\)
List the elements of \(R/I\) and \(R/J\text{.}\)
What happens to \(x^2\) in \(R\) when you pass to the quotient ring \(R/I\text{?}\) How about \(x^3\) as you pass from \(R\) to \(R/J\text{?}\)
In view of your answer to the previous question, how does \(x\) behave as you “mod out” by \(I\) and \(J\text{?}\)
Build addition and multiplication tables for one of \(R/I\) and \(R/J\text{.}\)
Exploration 1.6.14.
One of the most useful connections made in high school algebra is the connection between a function \(f\) (in particular, a polynomial function) and its graph. We may extend this notion to ideals via the concept of a zero set as follows.
Let \(F\) be a field and \(R = F[x,y]\) with \(I\subseteq R\) a nonzero ideal. We define the zero set of \(I\text{,}\) denoted \(Z(I)\text{,}\) as the set of all points \((a,b)\in F^2\) for which \(f(a,b)=0\) for all \(f\in I\text{.}\)
Suppose \(I = \ideal{f_1, f_2, \ldots, f_n}\text{.}\) Prove that \((a,b)\in Z(I)\) if and only if \(f_j(a,b) = 0\) for each \(j\in \set{1,\ldots, n}\text{.}\) Thus, \(Z(I)\) can be determined entirely by examining the generators of \(I\text{.}\)
Describe \(Z(I)\) given \(I = \ideal{y-x^2}\text{.}\)
(Challenge) Given \(I = \ideal{y-x^2}\) and \(J = \ideal{y-x-2}\text{,}\) describe \(Z(I+J)\) and \(Z(I\cap J)\text{.}\)
Given \(I=\ideal{y-x^2}\text{,}\) describe the relationship between the variables \(x\) and \(y\) in the quotient \(R/I\text{.}\) In what way have we restricted our polynomial “inputs” to the parabola \(y = x^2\text{?}\)
