Reduced ring

A reduced ring is a ring containing no nilpotent elements. ( Nilpotent elements result multiplied by itself eventually zero. ) Reduced rings play a role in commutative algebra and algebraic geometry, which are branches of mathematics. A reduced diagram is a diagram which blades are reduced.

This article deals with commutative algebra. In particular, all rings considered are commutative and have an identity element. Ring homomorphisms form elements from one to one elements. For more details see Commutative Algebra.

Definitions

Reduced ring

Is a ring, it is a reduced ring if all

This is equivalent to:

• The nilradical of the ring is the zero ideal:

• For all true:

Reduced Ideal

An ideal of a ring is a reduced ideal if:

Reduced scheme

A schema is reduced if for every open set of the ring contains no nilpotent elements. This is equivalent to saying that for all the local rings ( blades ):

Are reduced.

Properties

• Is noetherian, then:

• Minimalism is a local property:

Examples

• And all polynomial rings over fields are reduced.
• The ring is reduced.
• Each divisors of zero ring is reduced.
• Contains the nilpotent element, that is not reduced.
• The ring is not reduced, it contains the nilpotent element.
• A schema is exactly then integer if it is irreducible and reduced.