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A special kind of subring of a ring R is the subring generated by a subset X, which is defined as the intersection of all subrings of R containing X. [3] The subring generated by X is also the set of all linear combinations with integer coefficients of elements of X , including the additive identity ("empty combination") and multiplicative ...
An intersection of subrings is a subring. Given a subset E of R, the smallest subring of R containing E is the intersection of all subrings of R containing E, and it is called the subring generated by E. For a ring R, the smallest subring of R is called the characteristic subring of R. It can be generated through addition of copies of 1 and −1.
is called the fixed subring or, more traditionally, the ring of invariants under G. If S is a set of automorphisms of R, the elements of R that are fixed by the elements of S form the ring of invariants under the group generated by S. In particular, the fixed-point subring of an automorphism f is the ring of invariants of the cyclic group ...
For example, we can take the subring of complex numbers of the form +, with and integers. [4] The maximal order question can be examined at a local field level. This technique is applied in algebraic number theory and modular representation theory.
One defines the ring of integers of a non-archimedean local field F as the set of all elements of F with absolute value ≤ 1; this is a ring because of the strong triangle inequality. [12] If F is the completion of an algebraic number field, its ring of integers is the completion of the latter's ring of integers. The ring of integers of an ...
Central to the development of these subjects were the rings of integers in algebraic number fields and algebraic function fields, and the rings of polynomials in two or more variables. Noncommutative ring theory began with attempts to extend the complex numbers to various hypercomplex number systems.
The ring / of integers modulo n has characteristic n. If R is a subring of S , then R and S have the same characteristic. For example, if p is prime and q ( X ) is an irreducible polynomial with coefficients in the field F p {\displaystyle \mathbb {F} _{p}} with p elements, then the quotient ring F p [ X ] / ( q ( X ) ) {\displaystyle \mathbb ...
It is the integers localized at 2. More generally, given any commutative ring R and any prime ideal P of R , the localization of R at P is local; the maximal ideal is the ideal generated by P in this localization; that is, the maximal ideal consists of all elements a / s with a ∈ P and s ∈ R - P .