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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 ...
Finitely generated modules over the ring of integers Z coincide with the finitely generated abelian groups. These are completely classified by the structure theorem, taking Z as the principal ideal domain. Finitely generated (say left) modules over a division ring are precisely finite dimensional vector spaces (over the division ring).
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 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 ...
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 concept of a module also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers. [ 1 ] Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operations of addition between elements of the ring or module and is compatible ...
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.
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.