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V is the symmetry group of this cross: flipping it horizontally (a) or vertically (b) or both (ab) leaves it unchanged.A quarter-turn changes it. In two dimensions, the Klein four-group is the symmetry group of a rhombus and of rectangles that are not squares, the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180° rotation.
An extension of A by B is called split if it is equivalent to the trivial extension. There is a one-to-one correspondence between equivalence classes of extensions of A by B and elements of Ext 1 R (A, B). [9] The trivial extension corresponds to the zero element of Ext 1 R (A, B).
The function f : Z → Z/nZ, defined by f(a) = [a] n = a mod n is a surjective ring homomorphism with kernel nZ (see modular arithmetic). The complex conjugation C → C is a ring homomorphism (this is an example of a ring automorphism). For a ring R of prime characteristic p, R → R, x → x p is a ring endomorphism called the Frobenius ...
An element a of Z/nZ has a multiplicative inverse (that is, it is a unit) if it is coprime to n. In particular, if n is prime, a has a multiplicative inverse if it is not zero (modulo n). Thus Z/nZ is a field if and only if n is prime. Bézout's identity asserts that a and n are coprime if and only if there exist integers s and t such that
An elementary embedding of a structure N into a structure M of the same signature σ is a map h: N → M such that for every first-order σ-formula φ(x 1, …, x n) and all elements a 1, …, a n of N,
Monoid schemes can be turned into ring-theoretic schemes by means of a base extension functor – ⊗ F 1 Z that sends the monoid A to the Z‑module (i.e. ring) Z[A] / 0 A , and a monoid homomorphism f : A → B extends to a ring homomorphism f Z : A ⊗ F 1 Z → B ⊗ F 1 Z that is linear as a Z‑module homomorphism. The base extension of ...
Given a commutative ring A, an A-extension or an extension of an A-algebra is defined in the same way by replacing "ring" with "algebra over A" and "abelian groups" with "A-modules". An extension is said to be trivial or to split if ϕ {\displaystyle \phi } splits; i.e., ϕ {\displaystyle \phi } admits a section that is a ring homomorphism [ 2 ...
If d is a divisor of n, then the number of elements in Z/nZ which have order d is φ(d), and the number of elements whose order divides d is exactly d. If G is a finite group in which, for each n > 0, G contains at most n elements of order dividing n, then G must be cyclic. [note 2] The order of an element m in Z/nZ is n/gcd(n,m).