ground fields and rings
One commonality is generally found for the use of ground ring or field: the result is a unitial subring of the original. Outside of this requirement, the constraints are specific to context.
Given a ring with a 1, let be the subgroup of generated by under addition. This is consequently a subring of of the same characteristic as . Thus is it isomorphic to where is the characteristic of . This is the smallest unital subring of and so rightfully may be called the ground or base ring of .
When the characteristic of is prime, and so it may be called the ground field of .
Given a set of matrices , the ground ring is commonly the ring , and if required as a subring of then it is taken as the set of all scalar matrices.
Given a field and a set of field automorphisms , the ground/base field in this context is the fixed field (http://planetmath.org/Fixed) of the automorphisms. That is, the largest subfield of which is pointwise fixed by each . Since a field automorphism must fix the prime subfield, this definition always produces a field containing the prime subfield.
|Title||ground fields and rings|
|Date of creation||2013-03-22 15:54:22|
|Last modified on||2013-03-22 15:54:22|
|Last modified by||Algeboy (12884)|