--- title: "Ideals and Quotient Rings" description: "The ring counterpart of a normal subgroup is an ideal, and quotienting by it declares its elements to be zero. Whether an ideal is prime or maximal is read directly off the ring the quotient produces." canonical: https://lightmysky.com/learn/mathematics/ideals-and-quotient-rings-mt_uQBA9E1JIP source: https://lightmysky.com/learn/mathematics/ideals-and-quotient-rings-mt_uQBA9E1JIP.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Ideals and Quotient Rings The ring counterpart of a normal subgroup is an ideal, and quotienting by it declares its elements to be zero. Whether an ideal is prime or maximal is read directly off the ring the quotient produces. Subject: Mathematics · Area: Abstract Algebra · Ages 22 to 23 Page: https://lightmysky.com/learn/mathematics/ideals-and-quotient-rings-mt_uQBA9E1JIP ## Ready when they can - Check whether a subset of a ring is an ideal - Describe the quotient of a polynomial ring by a principal ideal - Distinguish prime from maximal in terms of the quotient ring ## Lesson: Ideals and what quotients do You work inside a ring, where you can add, subtract, and multiply. A subset I is an ideal when it passes two tests: differences stay inside, and multiplying any element of I by any element of the ring stays inside. You can check this on integers: the even integers and the multiples of 6 pass both tests, while the odd integers fail subtraction and the set containing 0, 1, and minus 1 fails absorption. **Example.** You meet the same two tests in two useful facts. The kernel of a ring homomorphism, the inputs sent to zero, is always an ideal of the source ring. And in integers, the ideal generated by 12 and 18 is the set of combinations 12s plus 18t, which equals the multiples of their greatest common divisor, so (12, 18) is the same as (6). You quotient by an ideal by declaring everything in it to be zero, so two elements count as the same when their difference lies in the ideal. For a field F and a degree 3 polynomial p, every class in F[x] divided by (p) has a unique remainder a plus bx plus cx squared, which takes three coefficients. For rationals, Q[x] divided by (x squared plus 1) forces x squared to equal minus 1, so each class looks like a plus bx and x plays the role of the imaginary unit. **Tip.** You read the ideal straight off its quotient. A maximal ideal is one whose quotient is a field, so in integers (n) is maximal exactly when n is prime, since Z divided by (5) is a field but Z divided by (6) is not. A prime ideal is one whose quotient has no zero divisors, so P is prime exactly when R divided by P is a domain. **Recap.** You test ideals by subtraction and absorption, and you read maximal and prime straight off the quotient ring. ## Practice 16 questions on this page, each with its working shown. ## Needs first - [Rings, Fields and Their First Properties](https://lightmysky.com/learn/mathematics/rings-fields-and-their-first-properties-mt_BmhIook13o) ## Opens up - [Polynomial Rings, Irreducibility and Unique Factorisation](https://lightmysky.com/learn/mathematics/polynomial-rings-irreducibility-and-unique-factorisation-mt_m2G9PIHj6y)