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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 ...
For example, the ring of integers is a subring of the field of real numbers and also a subring of the ring of polynomials [] (in both cases, contains 1, which is the multiplicative identity of the larger rings).
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 ...
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.
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 ...
The ring of integers ... For example, the ring ... it is a subring of . The ring of p-adic integers is an integral domain. The ...
An explicit example is the ring of integers Z, a Euclidean domain. All regular local rings are integrally closed as well. A ring whose localizations at all prime ideals are integrally closed domains is a normal ring .
For an example more geometrical in nature, take the ring R = {f/g : f, g polynomials in R[X] and g(0) ≠ 0}, considered as a subring of the field of rational functions R(X) in the variable X. R can be identified with the ring of all real-valued rational functions defined (i.e. finite) in a neighborhood of 0 on the real axis (with the ...