11.2 Properties of complete rings

Ring completions extend our understanding of algebraic structures. They allow us to work with infinite series and limits in rings, much like we do with real numbers. This process preserves important properties like being Noetherian or local.

Completions also maintain a close relationship with the original ring's ideal structure. Prime ideals, in particular, correspond neatly between a ring and its completion, preserving crucial algebraic information.

Completion of Rings

Uniqueness of ring completion

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• I-adic topology induced by powers of ideal I forms basis for open sets
• Completion of ring R with respect to I-adic topology constructed as inverse limit of R/I^n as n approaches infinity
• Uniqueness theorem states completion of ring R unique up to isomorphism
• Proof demonstrates uniqueness by constructing isomorphism between any two completions R' and R''
• Unique homomorphism φ: R' → R'' shown to be bijective establishes isomorphism

Noetherian property in completions

• Noetherian rings characterized by stabilizing ascending chains of ideals
• Completion of Noetherian ring R denoted as $\hat{R}$
• Proof strategy establishes correspondence between ideals in $\hat{R}$ and R
• Key steps involve considering ideal J in $\hat{R}$, defining its contraction J ∩ R
• Equality (J ∩ R)$\hat{R}$ = J leveraged with Noetherian property of R
• Noetherian property of R extends to $\hat{R}$ through this correspondence

Local Rings and Prime Ideals

Completion of local rings

• Local rings defined by unique maximal ideal
• Completion of local ring (R, m) denoted $\hat{R}$, uses m-adic topology
• Proof shows $\hat{R}$ is local by identifying unique maximal ideal
• Completeness of $\hat{R}$ demonstrated using definition of completion
• Process maintains local ring properties while achieving completeness

Prime ideals in rings vs completions

• Prime ideals in R defined by ab ∈ P implies a ∈ P or b ∈ P
• Prime ideals in $\hat{R}$ correspond to those in R
• Extension I$\hat{R}$ and contraction J ∩ R operations connect ideals between R and $\hat{R}$
• Bijection established between prime ideals of R containing I and prime ideals of $\hat{R}$
• Height of prime ideals preserved in this correspondence
• Study of prime ideals simplified by reducing to local case through completion