Introduction to Commutative Algebra

Atiyah, M. F., & Ian G. Macdonald, I. t. C. A. (1969). Introduction to Commutative Algebra. Addison-Wesley.

Type

Book chapter

Intellectual base

Textbook

Year

1969

ISBN

9780201003611

Link

https://www.routledge.com/Introduction-To-Commutative-Algebra/Atiyah-Macdonald/p/book/9780813345444

Cited by

10 citations across 10 artifacts.

Domain-specific

• Completion of a ring
  • The entry locks ordinary ideal-adic completion of commutative rings; noncommutative, derived, uniform-space, and metric completions require separately typed definitions.
• Connected ring
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Field of fractions
  • Field of fractions is the smallest field containing an integral domain, constructed from equivalence classes of ratios of its elements with nonzero denominators.
• Free presentation
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Nilradical of a ring
  • … the distinction between constitutive identity and a convenient observable transfers from In k[ε]/(ε²), the nilradical is the ideal generated by ε, and the quotient by it is k. to An algebraist computes nilpotent generators or intersects minimal primes and distinguishes the nilradical from the Jacobson radical..
• Polynomial Ring
  • Multivariate rings \(R[x_1,\ldots,x_m]\) similarly use finite \(R\)-linear combinations of monomials.
• Principal Ideal
  • In polynomial rings over a field, one-variable ideals are principal, while multivariable polynomial rings generally have ideals needing several generators.
• Product of Rings
  • A finite decomposition \(R\cong R_1\times R_2\) corresponds to complementary central idempotents under standard hypotheses.
• Ring Homomorphism
  • They are the morphisms that organize rings into a category and reverse direction when affine geometric spaces are represented by coordinate rings.
• Total ring of fractions
  • The total quotient ring Q(R), or total ring of fractions, is the localization S⁻¹R where S consists of all regular elements of R; for an integral domain it is the ordinary field of fractions.

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