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The prime subring of a ring R is a subring of the center of R, which is isomorphic either to the ring of the integers or to the ring of the integers modulo n, where n is the smallest positive integer such that the sum of n copies of 1 equals 0.
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 ...
The set of all algebraic integers A is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers. The ring of integers of a number field K, denoted by O K, is the intersection of K and A: it can also be characterised as the maximal order of the field K. Each algebraic integer belongs to ...
Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings ...
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.
The ring of p-adic integers is an integral domain. The ring of formal power series of an integral domain is an integral domain. If U {\displaystyle U} is a connected open subset of the complex plane C {\displaystyle \mathbb {C} } , then the ring H ( U ) {\displaystyle {\mathcal {H}}(U)} consisting of all holomorphic functions is an integral domain.
The integers arranged on a number line. An integer is the number zero , ... The smallest field containing the integers as a subring is the field of rational numbers.