abeliano

Abeliano is the term used in several languages to describe mathematical objects characterized by commutativity, most often expressed as abelian in English. The name honors the Norwegian mathematician Niels Henrik Abel, whose work on the solvability of equations and properties of commutativity influenced the terminology and development of algebra.

An abelian structure generally means that the primary operation is commutative. In its standard form, an abelian group is a set equipped with a binary operation that is closed, associative, has an identity element, every element has an inverse, and satisfies a*b = b*a for all elements a and b. When the operation is written additively, the identity is denoted 0 and the inverse of a is -a.

Examples include the integers Z under addition, the real numbers R under addition, and the vector space concept of addition of vectors. Finite cyclic groups Z_n (integers modulo n) are also abelian. More generally, the additive structure of any ring or module is abelian.

Abelian groups are central to algebra. They support notions of homomorphisms, subgroups, quotients, and direct sums. They appear in classification theorems, such as the fundamental theorem of finitely generated abelian groups, which describes their structure as a product of cyclic groups. In category theory and homological algebra, the term abelian extends to abelian categories, where morphisms and kernels/cokernels satisfy exactness properties, and many algebraic objects (modules, sheaves) form abelian categories.