generated subring

Definition 1

Let M be a nonempty subset of a ring A. The intersectionMathworldPlanetmath of all subrings of A that include M is the smallest subring of A that includes M. It is called the subring generated by M and is denoted by M.

The subring generated by M is formed by finite sums of monomials of the form :


Of particular interest is the subring generated by a family of subrings E={Ai|iI}. It is the ring R formed by finite sums of monomials of the form:


If A,B are rings, the subring generated by AB is also denoted by AB.
In the case when Ai are fields included in a larger field A then the set of all quotientsPlanetmathPlanetmath of elements of R ( the quotient field of R) is the composite field iIAi of the family E. In other words, it is the subfield generated by iIAi. The notation iIAi comes from the fact that the family of all subfields of a field forms a complete latticeMathworldPlanetmath.
The of fields is defined only when the respective fields are all included in a larger field.

Title generated subring
Canonical name GeneratedSubring
Date of creation 2013-03-22 16:57:27
Last modified on 2013-03-22 16:57:27
Owner polarbear (3475)
Last modified by polarbear (3475)
Numerical id 9
Author polarbear (3475)
Entry type Definition
Classification msc 20-00
Classification msc 13-00
Classification msc 16-00
Related topic RingAdjunction
Defines subring generated by
Defines monomials in rings