Below you will find example sentences with "boolean ring". The examples show how this phrase is used in real sentences and which words often surround it.
Boolean Ring in a sentence
About this phrase
- Belongs to the word: ring
Example types with boolean ring
Below, the examples are grouped by length and sentence type:
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)
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.