Below you will find example sentences with "boolean ring". The examples show how this phrase is used in natural context and which words often surround it.

Boolean Ring in a sentence

Corpus data

  • Displayed example sentences: 8
  • Discovered as a combination around: ring
  • Corpus frequency in the collocation scan: 7
  • Phrase length: 2 words
  • Average sentence length: 24.3 words

Words in this phrase

boolean ring

Meanings of words in this phrase

  • boolean can mean: Of or pertaining to the work of George Boole. Rare word
  • ring can mean: A solid object in the shape of a circle. Frequent word

Sentence profile

  • Phrase position: 2 start, 4 middle, 2 end
  • Sentence types: 8 statements, 0 questions, 0 exclamations

Usage patterns

These patterns show short text windows around the phrase in the example sentences:

again a boolean ring 1× in the boolean ring a the 1× into the boolean ring 1× is a boolean ring 1× mean a boolean ring possibly without 1× mean a boolean ring with an 1× of a boolean ring is a 1× of any boolean ring r modulo 1×

Sentence openings and endings

Common sentence openings:

by stone s 1× furthermore a subset 1× historically the term 1× more generally with 1× moreover these notions 1× the existence of 1×

Common sentence endings:

a boolean ring 2× a boolean algebra 1× is a filter 1× the boolean algebra 1× the boolean ring 1× with an identity 1×

Context words

These words often appear near this phrase in the example sentences:

ideal 10× algebra 4× field 4× identity 3× prime 3× maximal 3× order 3× used 2× mean 2× two 2× elements 2× sets 2×

Related phrases

engagement ring ring road ring video wedding ring diamond ring

Phrases around this combination

Other phrases discovered around ring:

ring road 11× ring video 8× wedding ring 8×

Just before it in the phrase index:

book value book titled book publishing book adaptation bonus points bonus bets

Just after it in the phrase index:

boot camps border controls border crossing border dispute border guards border patrol

Other phrases with the same pattern

Same ending:

engagement ring wedding ring diamond ring

Corpus analysis

  • The phrase "boolean ring" has 2 words and usually appears in the middle in these examples. The average sentence has 24.3 words and is mostly made up of statements.
  • Around this phrase, patterns and context words such as again a boolean ring, in the boolean ring a the, ideal, algebra and field stand out.
  • In the phrase index, this combination connects with engagement ring, ring road, ring video, ring road, ring video and wedding ring, linking the page to nearby combinations.

Example types with boolean ring

This selection groups the examples by length and sentence type, making usage of the full phrase easier to scan:

Short example sentences

Thus every Boolean ring becomes a Boolean algebra. (8 words)

More generally with these operations any field of sets is a Boolean ring. (13 words)

The quotient ring R/I of any Boolean ring R modulo any ideal I is again a Boolean ring. (19 words)

Longer example sentences

The existence of the identity is necessary to consider the ring as an algebra over the field of two elements : otherwise there cannot be a (unital) ring homomorphism of the field of two elements into the Boolean ring. (38 words)

Furthermore, a subset of a Boolean ring is a ring ideal (prime ring ideal, maximal ring ideal) if and only if it is an order ideal (prime order ideal, maximal order ideal) of the Boolean algebra. (36 words)

Historically, the term "Boolean ring" has been used to mean a "Boolean ring possibly without an identity", and "Boolean algebra" has been used to mean a Boolean ring with an identity. (31 words)

Example sentences (8)

Historically, the term "Boolean ring" has been used to mean a "Boolean ring possibly without an identity", and "Boolean algebra" has been used to mean a Boolean ring with an identity.

Furthermore, a subset of a Boolean ring is a ring ideal (prime ring ideal, maximal ring ideal) if and only if it is an order ideal (prime order ideal, maximal order ideal) of the Boolean algebra.

The quotient ring R/I of any Boolean ring R modulo any ideal I is again a Boolean ring.

Thus every Boolean ring becomes a Boolean algebra.

The existence of the identity is necessary to consider the ring as an algebra over the field of two elements : otherwise there cannot be a (unital) ring homomorphism of the field of two elements into the Boolean ring.

By Stone's representation theorem every Boolean ring is isomorphic to a field of sets (treated as a ring with these operations).

Moreover, these notions coincide with ring theoretic ones of prime ideal and maximal ideal in the Boolean ring A. The dual of an ideal is a filter.

More generally with these operations any field of sets is a Boolean ring.