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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 ...
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.
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 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 .
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 question of when this happens is rather subtle: for example, for the localization of k[x, y, z]/(x 2 + y 3 + z 5) at the prime ideal (x, y, z), both the local ring and its completion are UFDs, but in the apparently similar example of the localization of k[x, y, z]/(x 2 + y 3 + z 7) at the prime ideal (x, y, z) the local ring is a UFD but ...
For example, if R is a principal ideal domain, then Pic(R) vanishes. In algebraic number theory, R will be taken to be the ring of integers, which is Dedekind and thus regular. It follows that Pic(R) is a finite group (finiteness of class number) that measures the deviation of the ring of integers from being a PID.
Thus, the zero-product property holds for any subring of a skew field. If is a prime number, then the ring of integers modulo has the zero-product property (in fact, it is a field). The Gaussian integers are an integral domain because they are a subring of the complex numbers.