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A special kind of subring of a ring R is the subring generated by a subset X, which is defined as the intersection of all subrings of R containing X. [3] The subring generated by X is also the set of all linear combinations with integer coefficients of elements of X , including the additive identity ("empty combination") and multiplicative ...
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.
In algebraic number theory there are examples for any other than the rational field of proper subrings of the ring of integers that are also orders. For example, in the field extension = of Gaussian rationals over , the integral closure of is the ring of Gaussian integers [] and so this is the unique maximal-order: all other orders in are ...
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 ...
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 ...
In this case, the conductor is non-zero. This applies in particular when B is the ring of integers in an algebraic number field and A is an order (a subring for which B /A is finite). The conductor is also an ideal of B, because, for any b in B and any a in (/), baB ⊆ aB ⊆ A.
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.
For example, [] is the smallest subring of C containing all the integers and ; it consists of all numbers of the form +, where m and n are arbitrary integers. Another example: Z [ 1 / 2 ] {\displaystyle \mathbf {Z} [1/2]} is the subring of Q consisting of all rational numbers whose denominator is a power of 2 .