Abstract Algebra: Theory and Applications
Thomas W. Judson · Free textbook
Read §§16.1 and 16.3 for rings and ideals, then §17.1 for polynomial rings; repeat the three-row reduction.
Open the sourceRings: a language for combining constraints
Relations let us calculate without pretending we have ordinary numbers.
Prerequisites: 02 · Why enlarge the number system?, 12 · Counting without listing everything, 21 · Matroid polytopes: selections become points
Understand
Objectives
- State the ideal relation and the surviving degree-one class.
- Expand before applying quotient relations.
- Explain why cancellation by x is invalid.
Start with something you can see
In ordinary real numbers, a square vanishes only when its base vanishes. What if a calculation explicitly imposes x²=0? Predict (1+x)³ before reducing it. First multiply as a polynomial, then use the stated relation; these are separate steps.
Give the idea a precise name
A ring permits addition and multiplication with distributivity. The polynomial ring ℝ[x] contains formal polynomials. An ideal is closed under addition and under multiplication by arbitrary ring elements. The quotient ℝ[x]/(x²) identifies polynomials whose difference is a multiple of x²; each class has one representative a+bx. This is not the real-number system.
ℝ[x]/(x²): x²=0, while x≠0
The ideal (x²) consists of x² times arbitrary polynomials, so x itself is not in that ideal. Every xⁿ for n≥2 vanishes in the quotient. Grading records degrees: constants have degree zero and x has degree one; degree-two products vanish here. We may subtract and distribute, but cannot cancel x from x·x=0 because x has no multiplicative inverse.
Work one small world
Reduce (1+x)³ step by step in the quotient. Each row gives the rule used.
| Stage | Expression | Rule |
|---|---|---|
| Start | (1+x)³ | multiply three factors |
| Expand | 1+3x+3x²+x³ | distributivity, collect like terms |
| Reduce | 1+3x | x²=0 and x³=x·x²=0 |
Similarly, (2+x)(3−x)=6+x−x² becomes 6+x. These equalities hold in the quotient, not as polynomial identities in ℝ[x].
Counterexample to cancellation by a nonzero element: x·x=0=x·0 in ℝ[x]/(x²), but x≠0. Canceling x would give the false conclusion x=0. Also (1+x)(1−x)=1−x²=1 in this ring, although the unreduced polynomial is not 1.
THE BRIDGEThe next ring has six degree-one generators and two sorts of relations, but uses the same expand-then-reduce discipline.
WHERE THIS IDEA STOPSA polynomial equality in a quotient need not hold in the ordinary polynomial ring.
Experiment
Cover the Reduce row and multiply (1+x)³ on paper. Reveal the quotient relation only after listing the powers that occur.
Which rules of real-number arithmetic are you assuming without checking whether this ring has them?
Check & explain
What is the coefficient of x in (2+x)(3−x) modulo x²?
Does x²=0 in ℝ[x]/(x²) imply x=0?
- Yes, by cancellation
- No; x is a nonzero nilpotent class
- The quotient has no multiplication
Teach back
Reduce (1+2x)³ and explain which product terms disappear. Then name the invalid step in a proposed cancellation of x·x=x·0.
- Expand using distributivity.
- Apply only the imposed ideal relation.
- Distinguish a nonzero nilpotent from an invertible real number.
Read deeper
Lecture Notes: Modern Algebra
MIT OpenCourseWare · Course notes · MIT OCW
Use modern algebra lecture 16 for ring homomorphisms and ideals; then work the quotient example here.
Open the sourceThe what, why, and how of quotient rings — Rings and Fields 7
MathMajor · Lecture · 58 min
Watch the full 58-minute lecture as optional quotient-ring background; the table is this lesson’s calculation.
Open the source