7.2 Properties of Noetherian rings and modules

Noetherian modules and rings are key players in algebra. They have special properties that make them easier to work with, like having well-behaved submodules and quotients. These structures pop up all over math, from basic number theory to advanced geometry.

Noetherian rings are particularly useful because their ideals are always finitely generated. This property extends to polynomial rings, thanks to Hilbert's Basis Theorem. It's a powerful tool that helps us understand more complex algebraic structures.

Noetherian Modules and Their Properties

Properties of Noetherian submodules

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• Definition of Noetherian modules entails every ascending chain of submodules stabilizes after finite steps (M1 ⊆ M2 ⊆ ... ⊆ Mn = Mn+1 = ...)
  • Equivalent characterization posits every non-empty set of submodules contains a maximal element under inclusion
• Submodules of Noetherian modules inherit Noetherian property
  • Consider N as submodule of Noetherian M
  • Ascending chain in N (N1 ⊆ N2 ⊆ ...) forms ascending chain in M
  • Chain stabilizes in M due to Noetherian property, thus stabilizes in N
• Quotient modules of Noetherian modules preserve Noetherian structure
  • For Noetherian M and submodule N, examine quotient M/N
  • Correspondence theorem links submodules of M/N to submodules of M containing N
  • Ascending chain condition transfers through this correspondence

Noetherian nature of finite modules

• Finite direct sums of Noetherian modules maintain Noetherian property
  • Consider $M = M_1 \oplus M_2 \oplus ... \oplus M_n$ with Noetherian $M_i$
  • Proof employs induction on n
  • Base case (n = 1) trivial, inductive step assumes true for n-1, proves for n
• Finite products of Noetherian modules mirror direct sum behavior
  • Finite direct product isomorphic to direct sum
  • Noetherian nature follows from direct sum result
• Infinite direct sums/products may lose Noetherian property
  • Counterexample: $\mathbb{Z} \oplus \mathbb{Z} \oplus ...$

Noetherian Rings and Their Extensions

Noetherian rings in polynomials

• Hilbert's Basis Theorem extends Noetherian property to polynomial rings
  • R Noetherian implies R[x] Noetherian
  • Proof uses induction on variable count
  • Ideals in R[x] correspond to R-submodules of R[x]
• Power series rings over Noetherian rings inherit Noetherian structure
  • R Noetherian leads to R[[x]] Noetherian
  • Proof parallels Hilbert's Basis Theorem, adapts for infinite degree terms
• Multivariate polynomial and power series rings preserve Noetherian property
  • R[x1, ..., xn] and R[[x1, ..., xn]] Noetherian when R Noetherian

Applications of Noetherian structures

• Ascending chain condition for ideals characterizes Noetherian rings
  • Every ascending chain of ideals stabilizes (I1 ⊆ I2 ⊆ ... ⊆ In = In+1 = ...)
  • Proves finiteness properties in ring theory
• Noetherian rings feature finitely generated ideals
  • Ring Noetherian if and only if every ideal finitely generated
  • Useful in proving other ring properties (primary decomposition)
• Quotient rings of Noetherian rings remain Noetherian
  • R Noetherian implies R/I Noetherian for any ideal I
  • Applies to factor rings and homomorphic images
• Noetherian property links rings and modules
  • Ring R Noetherian equivalent to R Noetherian as R-module
  • Bridges results between ring theory and module theory
• Algebraic geometry utilizes Noetherian rings
  • Noetherian rings crucial in affine variety study
  • Connects to Hilbert's Nullstellensatz (zero locus of ideals)
• Krull's intersection theorem applies to Noetherian rings
  • $\cap_{n=1}^{\infty} I^n = (0)$ for proper ideal I in Noetherian local ring
  • Important in local ring theory (completion, depth)
• Primary decomposition exists in Noetherian rings
  • Every ideal admits primary decomposition
  • Minimal primary decomposition unique
  • Aids in ideal structure analysis (associated primes, radical)