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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.
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 ...
2. A central algebra is an associative algebra over the centre. 3. A central simple algebra is a central algebra that is also a simple ring. centralizer 1. The centralizer of a subset S of a ring is the subring of the ring consisting of the elements commuting with the elements of S. For example, the centralizer of the ring itself is the centre ...
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.
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 .
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 .
An abelian group is the same thing as a module over the ring of integers, which is the initial object in the category of rings. In a similar fashion, if R is any commutative ring , the endomorphisms of an R -module form an algebra over R by the same axioms and derivation.