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The intersection of all subrings of a ring R is a subring that may be called the prime subring of R by analogy with prime fields. The prime subring of a ring R is a subring of the center of R , which is isomorphic either to the ring Z {\displaystyle \mathbb {Z} } of the integers or to the ring of the integers modulo n , where n is the smallest ...
The subring consisting of elements with finite support is called the group ring of G (which makes sense even if G is not commutative). If G is the ring of integers, then we recover the previous example (by identifying f with the series whose n th coefficient is f(n
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 ...
Finitely generated modules over the ring of integers Z coincide with the finitely generated abelian groups. These are completely classified by the structure theorem, taking Z as the principal ideal domain. Finitely generated (say left) modules over a division ring are precisely finite dimensional vector spaces (over the division ring).
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.
In mathematics, Galois rings are a type of finite commutative rings which generalize both the finite fields and the rings of integers modulo a prime power. A Galois ring is constructed from the ring Z / p n Z {\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} } similar to how a finite field F p r {\displaystyle \mathbb {F} _{p^{r}}} is constructed ...
In particular, the integers (also see Fundamental theorem of arithmetic), the Gaussian integers and the Eisenstein integers are UFDs. If R is a UFD, then so is R[X], the ring of polynomials with coefficients in R. Unless R is a field, R[X] is not a principal ideal domain. By induction, a polynomial ring in any number of variables over any UFD ...
Likewise, any subring of a Boolean ring is a Boolean ring. Any localization RS −1 of a Boolean ring R by a set S ⊆ R is a Boolean ring, since every element in the localization is idempotent. The maximal ring of quotients Q(R) (in the sense of Utumi and Lambek) of a Boolean ring R is a Boolean ring, since every partial endomorphism is ...