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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 ...
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 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
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 ...
Let be a group, written multiplicatively, and let be a ring. The group ring of over , which we will denote by [], or simply , is the set of mappings : of finite support (() is nonzero for only finitely many elements ), where the module scalar product of a scalar in and a mapping is defined as the mapping (), and the module group sum of two mappings and is defined as the mapping () + ().
An element x in K is said to be almost integral over A if the subring A[x] of K generated by A and x is a fractional ideal of A; that is, if there is a nonzero such that for all . Then A is said to be completely integrally closed if every almost integral element of K is contained in A .
The Gaussian integers are the set [1] [] = {+,}, =In other words, a Gaussian integer is a complex number such that its real and imaginary parts are both integers.Since the Gaussian integers are closed under addition and multiplication, they form a commutative ring, which is a subring of the field of complex numbers.
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.