# OpenMath Content Dictionary: ring4

Canonical URL:
http://www.openmath.org/cd/ring4.ocd
CD Base:
http://www.openmath.org/cd
CD File:
ring4.ocd
CD as XML Encoded OpenMath:
ring4.omcd
Defines:
is_domain, is_field, is_maximal_ideal, is_prime_ideal, is_zero_divisor
Date:
2004-06-01
Version:
1 (Revision 1)
Review Date:
2006-06-01
Status:
experimental

A CD of functions for further basic properties of rings

Written by Arjeh M. Cohen 2004-02-25


## is_maximal_ideal

Description:

The binary boolean function whose value is true iff the second argument is a maximal ideal of the first.

Signatures:
sts

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## is_prime_ideal

Description:

The binary boolean function whose value is true iff the second argument is a prime ideal of the first.

Signatures:
sts

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## is_domain

Description:

This symbol represents a boolean unary function. The argument is a ring R. When evaluated on R, the function returns true if R is a domain and false otherwise. A domain is a commutative ring without zero divisors.

Signatures:
sts

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## is_field

Description:

This is unary boolean function whose argument should be a ring R. The value is true if and only if the ring is commutative and every nonzero element has a multiplicative inverse.

Commented Mathematical property (CMP):
If is_commutative(G) and for all a in carrier(G) there is b in carrier(G) such that a*b = identity(G), then is_field(G).
Formal Mathematical property (FMP):
$\mathrm{is_commutative}\left(G\right)\wedge \forall a.a\in \mathrm{carrier}\left(G\right)\wedge a\ne \mathrm{zero}\left(G\right)⇒\exists b.b\in \mathrm{carrier}\left(G\right)\wedge \left(\mathrm{multiplication}\left(G\right)\right)\left(a,b\right)=\mathrm{identity}\left(G\right)⇒\mathrm{is_field}\left(G\right)$
Signatures:
sts

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## is_zero_divisor

Description:

This symbol represents a boolean binary function. The first argument is a ring R, the second is an element x of R. When evaluated on R and x, the function returns true if x a zero divisor and nonzero in R.

Commented Mathematical property (CMP):
An element x of a ring R is a zero divisor if and only if it nonzero and there is a nonzero y in R such that x * y = 0 or y * x = 0.
Signatures:
sts

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