Commutative Algebra

Studying Commutative Algebra

Localization lets algebra inspect what survives near a chosen prime, while integral dependence controls how rings grow. Those two operations connect the early subject to geometry and number theory. Milne offers a short selection centered on ideals, Noetherian rings, fractions and integrality. Altman and Kleiman offer a more systematic term-sized development in ring and module language, which suits readers ready to keep tensor products and exact sequences active.

What to work on

• Test prime and maximal ideals using quotients.
• Construct a localization and track extension and contraction of ideals.
• Use the ascending-chain condition and Hilbert's basis theorem.
• Prove that a specified element is integral by finding a monic equation.

A starting text

A Primer of Commutative Algebra — James S. Milne. Read section 2, Ideals, as needed; then section 3, Noetherian rings, printed pages 9-13; section 5, Rings of fractions, pages 18-24; and section 6, Integral dependence, pages 25-29. Use section 14, The spectrum of a ring, only for the geometric continuation.

Practice

Localization and integrality exercises with selected solutions. Exercises in sections 2, 3, 5 and 6; consult Appendix A only for exercise numbers actually listed there.. Exercises provided; public solutions to selected numbered exercises are present in Appendix A, printed page 110. Full coverage is not claimed.

Try these before moving on

Try a proof or calculation before opening the hint. These solutions address the checks below, not the exercises in the linked textbooks.

1. Describe Z localized at the complement of (5), determine its units, and list its prime ideals with justification.

   A hint

   Denominators cannot be divisible by 5. Use prime ideals of the integers disjoint from the denominator set.

   Work through the solution

   Let $S=\mathbb Z\setminus(5)$. As a subring of $\mathbb Q$, $R=S^{-1}\mathbb Z=\{a/b:a,b\in\mathbb Z,\ 5\nmid b\}$. If $5\nmid a$, the inverse $b/a$ lies in $R$. If $5\mid a$, reduction modulo $5$ sends $a/b$ to $0$ in $\mathbb F_5$ and thus it cannot be a unit. Therefore the units are precisely the fractions whose numerators are also prime to $5$.

   Prime ideals of a localization correspond by extension and contraction to prime ideals disjoint from $S$. The primes of $\mathbb Z$ are $(0)$ and $(p)$ for positive primes $p$. The ideal $(0)$ is disjoint from $S$, as is $(5)$. For $p\ne5$, the element $p$ belongs to both $(p)$ and $S$. Hence the only primes of $R$ are $(0)$ and $5R$. The first is prime because $R$ is a domain; the second is maximal since $R/5R\cong\mathbb F_5$.

   If this was the sticking point:

   • Abstract Algebra & Galois Theory — Review quotient rings, prime ideals and extension/contraction under ring maps.

Already comfortable with this? Skip sections 2-3 if ideal correspondences and Noetherian proofs are routine. Skip localization and integrality only if you can do both checks and state the universal property of localization; familiarity with the notation S inverse A is insufficient.

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Reading choices revised 2026-10-17.

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