Reduced ring

A reduced ring is a ring that contains no other nilpotent elements apart from the zero element . (Nilpotent elements produce zero when raised to the power.) Reduced rings play a role in commutative algebra and algebraic geometry , which are sub-areas of mathematics. A reduced scheme is a scheme whose stems are reduced.

This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. Ring homomorphisms map single elements onto single elements. For more details, see Commutative Algebra .

Definitions

Reduced ring

If there is a ring, then it is a reduced ring, if for all ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle r\in R}$

${\displaystyle r^{n}=0\Leftrightarrow r=0}$

This is equivalent to:

• The Nile radical of the ring is the zero ideal:

${\displaystyle {\sqrt {(0)}}=(0)}$

• The following applies to all : ${\displaystyle r\in R}$

${\displaystyle r^{2}=0\Leftrightarrow r=0}$

Reduced ideal

An ideal of a ring is a reduced ideal if: ${\displaystyle I}$ ${\displaystyle R}$

${\displaystyle {\sqrt {I}}=I}$

Reduced scheme

A scheme is reduced if, for any open set, the ring does not contain any nilpotent elements. This is equivalent to the fact that for all the local rings (stalks): ${\displaystyle (X,{\mathcal {O}}_{X})}$ ${\displaystyle U\subset X}$ ${\displaystyle {\mathcal {O}}_{X}(U)}$ ${\displaystyle x\in X}$

${\displaystyle {\mathcal {R}}_{x}=\operatorname {lim} _{V\ni x}{\mathcal {F}}(V)}$

are reduced.

properties

• If noetherian then applies: ${\displaystyle R}$

${\displaystyle R}$is reduced is equivalent to the fact that in the primary decomposition of its zero ideal only prime ideals appear as primary components (the minimal prime ideals).

• Reducedness is a local characteristic :

A ring is reduced precisely when it is reduced for all maximum ideals. ${\displaystyle R}$ ${\displaystyle R_{m}}$

Examples

• ${\displaystyle \mathbb {Z} }$ and all polynomial rings over bodies are reduced.
• The ring is reduced. ${\displaystyle R/{\sqrt {(0)}}}$
• Every ring free of zero divisors is reduced.
• ${\displaystyle Z/4}$contains the nilpotent element , so it is not reduced. ${\displaystyle 2}$
• The ring is not reduced, it contains the nilpotent element . ${\displaystyle K[X]/(X^{2})}$ ${\displaystyle X}$
• A schema is of integrity if and only if it is irreducible and reduced.

literature

• Ernst Kunz : Introduction to Commutative Algebra and Algebraic Geometry , Vieweg (1980), ISBN 3-528-07246-6
• Atiyah , Macdonald : Introduction to Commutative Algebra , Addison-Wesley (1969), ISBN 0-2010-0361-9
• Brüske, Ischebeck, Vogel: Commutative Algebra , Bibliographisches Institut (1989), ISBN 978-3411140411
• H. Matsumura, Commutative algebra 1980 ISBN 0-8053-7026-9 .