7.1 Definition and characterizations of Noetherian rings

Noetherian rings are a key concept in commutative algebra. They're defined by the ascending chain condition on ideals, which means any chain of ideals eventually stops growing. This property has several equivalent characterizations.

One important feature of Noetherian rings is that all their ideals are finitely generated. This leads to powerful applications in algebra and geometry, including the famous Hilbert Basis Theorem for polynomial rings.

Definition and Characterizations of Noetherian Rings

Noetherian rings and characterizations

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• Noetherian ring defined as commutative ring $R$ satisfying ascending chain condition (ACC) on ideals
• ACC on ideals means any ascending chain $I_1 \subseteq I_2 \subseteq I_3 \subseteq \cdots$ eventually stabilizes at $I_n$ for some $n$
• Equivalent characterizations include maximal condition where every non-empty set of ideals has a maximal element
• Every ideal in $R$ must be finitely generated

Finite generation of ideals

• Noetherian ring implies all ideals finitely generated
  • For arbitrary ideal $I$, choose $a_1 \in I$, consider $(a_1) \subseteq I$
  • If $(a_1) \neq I$, choose $a_2 \in I \setminus (a_1)$, continue process
  • Chain $(a_1) \subseteq (a_1, a_2) \subseteq (a_1, a_2, a_3) \subseteq \cdots$ stabilizes by ACC
  • $I = (a_1, \ldots, a_n)$ for some $n$
• Finitely generated ideals imply Noetherian ring
  • For ascending chain $I_1 \subseteq I_2 \subseteq I_3 \subseteq \cdots$, let $J = \cup_{i=1}^{\infty} I_i$
  • $J$ finitely generated as $J = (a_1, \ldots, a_n)$
  • Each $a_i$ in some $I_{k_i}$, so $J \subseteq I_m$ where $m = \max\{k_1, \ldots, k_n\}$
  • $I_k = I_m$ for all $k \geq m$, proving ACC

Applications and Examples

Identification of Noetherian rings

• Noetherian rings include fields (ideals are (0) or entire field), principal ideal domains ($\mathbb{Z}$, $F[x]$ for $F$ a field)
• Non-Noetherian rings encompass polynomial rings with infinitely many variables over a field, ring of continuous functions on $\mathbb{R}$
• Determine Noetherian status by checking finite generation of ideals, attempting to construct infinite ascending chain, or using properties (quotients and finite extensions of Noetherian rings are Noetherian)

Noetherian rings vs Hilbert Basis Theorem

• Hilbert Basis Theorem states if $R$ is Noetherian, then $R[x]$ is also Noetherian
• Proof involves showing finite generation of ideal $I$ in $R[x]$ using leading coefficients, Noetherian property of $R$, and induction on polynomial degree
• Implies $F[x_1, \ldots, x_n]$ Noetherian for any field $F$ and finite $n$
• Polynomial rings in finitely many variables over Noetherian rings are Noetherian
• Theorem solved Gordan's problem in invariant theory and led to development of computational algebraic geometry