In algebra, the fixed-point subring R f {\displaystyle R^{f}} of an automorphism f of a ring R is the subring of the fixed points of f, that is, R f =...

f(g)=g\}.} Similarly, the fixed-point subring R f {\displaystyle R^{f}} of an automorphism f of a ring R is the subring of the fixed points of f, that is,...

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...

D (also called an "involution of the first kind") such that the fixed-point subring Dσ has dimension d(d + 1)/2; or equivalently, when d > 1 and char...

of powers of two. As well as forming a subring of the real numbers, the dyadic rational numbers form a subring of the 2-adic numbers, a system of numbers...

K} with discrete valuation ν {\displaystyle \nu } we can associate the subring O K := { x ∈ K ∣ ν ( x ) ≥ 0 } {\displaystyle {\mathcal {O}}_{K}:=\left\{x\in...

ring R = {f/g : f, g polynomials in R[X] and g(0) ≠ 0}, considered as a subring of the field of rational functions R(X) in the variable X. R can be identified...

an E∞-ring with an action of a finite group G and B = AhG its invariant subring. Then B → A (the map of B-algebras in E∞-sense) is said to be a G-Galois...

primarily as sets with operations. Algebraic structure Subobjects: subgroup, subring, subalgebra, submodule etc. Binary operation Closure of an operation Associative...

according to the permutation used. The invariants for this action form the subring of symmetric polynomials. If the indeterminates are X1, ..., Xn, then examples...

of the polynomial B. The obvious choice for such a subring is the centralizer Z of A, the subring of all matrices that commute with A; by definition A...

is prime. Prime avoidance lemma. If R is a commutative ring, and A is a subring (possibly without unity), and I1, ..., In is a collection of ideals of...

basic in quantum mechanics: D x − x D = 1. {\displaystyle Dx-xD=1.} The subring of operators that are polynomials in D with constant coefficients is, by...

include the nonassociative case). This amounts to finding the largest subring of O containing the units on which the expressions x*x (the norm of x)...

over its center or any subring lying in the center. In particular, any commutative ring is an algebra over any of its subrings. Other examples abound...

combination of these binomial coefficient polynomials. More generally, for any subring R of a characteristic 0 field K, a polynomial in K[t] takes values in R...

\,\ldots ,\,a_{n}^{-1}).} In particular, the diagonal matrices form a subring of the ring of all n-by-n matrices. Multiplying an n-by-n matrix A from...

primes ⁠ p {\displaystyle p} ⁠, x p {\displaystyle x_{p}} belongs to the subring Z p {\displaystyle \mathbb {Z} _{p}} of p-adic integers. Given any algebraic...

integers, which contains the ring of rationals with odd denominators as a subring. When using the "shortcut" definition of the Collatz map, it is known that...

the table structure itself The tables can have multiple dimensions, of fixed or variable lengths and are usually portable between computer platforms...

squared 2. The D4 root lattice is the dual of the F4 and is given by the subring of Hurwitz quaternions with even norm squared. Viewed as the 24 unit Hurwitz...

product of two such matrices is again of this form, these matrices form a subring of the ring of 2 × 2 matrices. A simple computation shows that the map...

multiplication. Ring theory is the study of rings, exploring concepts such as subrings, quotient rings, polynomial rings, and ideals as well as theorems such...

F(S ) be the smallest subobject of X that contains S, i.e. the subgroup, subring or subspace generated by S. For any subobject U of X, let G(U ) be the...

functions are analytic). Let G be a finite group and let Char(G) denote the subring of the ring of complex-valued class functions of G consisting of integer...

v k ∈ D {\displaystyle v_{1},\dots ,v_{k}\in D} . The set I(D) forms a subring and, in fact, an ideal in Ω(M). Furthermore, using the definition of the...

a subring of (that is, a ring contained in) K {\displaystyle K} . A field contains no zero divisors and this property is inherited by any subring, so...

are subgroups, submonoids, subrings, subfields, subalgebras of algebras over a field, or induced subgraphs. Shifting the point of view, the larger structure...

is a G-ring but not a J-2 ring and so not quasi-excellent. If R is the subring of the polynomial ring k[x1,x2,...] in infinitely many generators generated...

closed under addition and multiplication (except S0), but S∗ is; it is the subring of the rationals consisting of all dyadic fractions. There are infinite...