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
This is equivalent to:
- The Nile radical of the ring is the zero ideal:
- The following applies to all :
Reduced ideal
An ideal of a ring is a reduced ideal if:
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):
are reduced.
properties
- If noetherian then applies:
- 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.
Examples
- and all polynomial rings over bodies are reduced.
- The ring is reduced.
- Every ring free of zero divisors is reduced.
- contains the nilpotent element , so it is not reduced.
- The ring is not reduced, it contains the nilpotent element .
- 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 .