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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 ...
The prototypical example is the ring of integers with the two operations of addition and multiplication. The rational, real and complex numbers are commutative rings of a type called fields . A unital associative algebra over a commutative ring R is itself a ring as well as an R -module .
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 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 ...
forms a subring of the Hurwitz quaternions H. Hurwitz integers have the advantage over Lipschitz integers that it is possible to perform Euclidean division on them, obtaining a small remainder. Both the Hurwitz and Lipschitz quaternions are examples of noncommutative domains which are not division rings.
A subring of a matrix ring is again a matrix ring. Over a rng, one can form matrix rngs. When R is a commutative ring, the matrix ring M n (R) is an associative algebra over R, and may be called a matrix algebra. In this setting, if M is a matrix and r is in R, then the matrix rM is the matrix M with each of its entries multiplied by r.
The function |x-y|, supplemented by |0|=0, is the restriction of an absolute value defined on the field of fractions of the discrete valuation ring. A DVR is compact if and only if it is complete and its residue field R/M is a finite field. Examples of complete DVRs include the ring of p-adic integers and; the ring of formal power series over ...
All rings are rngs. A simple example of a rng that is not a ring is given by the even integers with the ordinary addition and multiplication of integers. Another example is given by the set of all 3-by-3 real matrices whose bottom row is zero. Both of these examples are instances of the general fact that every (one- or two-sided) ideal is a rng.