reduced ring
A ring is said to be a reduced ring if contains no non-zero nilpotent elements. In other words, implies for any .
Below are some examples of reduced rings.
-
•
A reduced ring is semiprime.
-
•
A ring is a domain (http://planetmath.org/CancellationRing) iff it is prime (http://planetmath.org/PrimeRing) and reduced.
-
•
A commutative semiprime ring is reduced. In particular, all integral domains and Boolean rings are reduced.
-
•
Assume that is commutative, and let be the set of all nilpotent elements. Then is an ideal of , and that is reduced (for if , then , so , and consequently , is nilpotent, or ).
An example of a reduced ring with zero-divisors is , with multiplication defined componentwise: . A ring of functions taking values in a reduced ring is also reduced.
Some prototypical examples of rings that are not reduced are , since , as well as any matrix ring over any ring; as illustrated by the instance below
Title | reduced ring |
---|---|
Canonical name | ReducedRing |
Date of creation | 2013-03-22 14:18:12 |
Last modified on | 2013-03-22 14:18:12 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 17 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 16N60 |
Synonym | nilpotent-free |