Concept
Free abelian group
Related concepts (28)
Smith normal form
In mathematics, the Smith normal form (sometimes abbreviated SNF) is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can be obtained from the original matrix by multiplying on the left and right by invertible square matrices. In particular, the integers are a PID, so one can always calculate the Smith normal form of an integer matrix.
Fundamental pair of periods
In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that defines a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and modular forms are defined. A fundamental pair of periods is a pair of complex numbers such that their ratio is not real. If considered as vectors in , the two are not collinear. The lattice generated by and is This lattice is also sometimes denoted as to make clear that it depends on and It is also sometimes denoted by or or simply by The two generators and are called the lattice basis.
Torsion group
In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which every element has finite order. The exponent of such a group, if it exists, is the least common multiple of the orders of the elements. For example, it follows from Lagrange's theorem that every finite group is periodic and it has an exponent dividing its order. Examples of infinite periodic groups include the additive group of the ring of polynomials over a finite field, and the quotient group of the rationals by the integers, as well as their direct summands, the Prüfer groups.
Free group
In mathematics, the free group FS over a given set S consists of all words that can be built from members of S, considering two words to be different unless their equality follows from the group axioms (e.g. st = suu−1t, but s ≠ t−1 for s,t,u ∈ S). The members of S are called generators of FS, and the number of generators is the rank of the free group. An arbitrary group G is called free if it is isomorphic to FS for some subset S of G, that is, if there is a subset S of G such that every element of G can be written in exactly one way as a product of finitely many elements of S and their inverses (disregarding trivial variations such as st = suu−1t).
Additive category
In mathematics, specifically in , an additive category is a C admitting all finitary biproducts. There are two equivalent definitions of an additive category: One as a category equipped with additional structure, and another as a category equipped with no extra structure but whose objects and morphisms satisfy certain equations. A category C is preadditive if all its hom-sets are abelian groups and composition of morphisms is bilinear; in other words, C is over the of abelian groups.
Chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the of each homomorphism is included in the kernel of the next. Associated to a chain complex is its homology, which describes how the images are included in the kernels. A cochain complex is similar to a chain complex, except that its homomorphisms are in the opposite direction. The homology of a cochain complex is called its cohomology.
Projective object
In , the notion of a projective object generalizes the notion of a projective module. Projective objects in are used in homological algebra. The dual notion of a projective object is that of an injective object. An in a category is projective if for any epimorphism and morphism , there is a morphism such that , i.e. the following diagram commutes: That is, every morphism factors through every epimorphism . If C is , i.e.
Universal coefficient theorem
In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space X, its integral homology groups: Hi(X; Z) completely determine its homology groups with coefficients in A, for any abelian group A: Hi(X; A) Here Hi might be the simplicial homology, or more generally the singular homology. The usual proof of this result is a pure piece of homological algebra about chain complexes of free abelian groups.