homomorfismien

Homomorfismen are maps between algebraic structures that preserve their defining operations. The term is used across different contexts, with group, ring, and module homomorphisms being the most common. For a group homomorphism f, the operation is preserved: f(xy) = f(x)f(y). For a ring homomorphism, both addition and multiplication are preserved: f(a+b) = f(a)+f(b) and f(ab) = f(a)f(b); unital ring homomorphisms additionally require f(1) = 1. For modules and vector spaces, a homomorphism is a linear map that preserves addition and scalar multiplication.

Examples illustrate the concept. The map f: Z → Z defined by f(n) = 2n is a group homomorphism from the additive group of integers to itself. The projection map from Z × Z onto the first coordinate is another group homomorphism. For rings, the reduction map π: Z → Z/2Z is a ring homomorphism. Linear maps between vector spaces are precisely the homomorphisms in the category of vector spaces.

Key notions associated with homomorphisms include the kernel and the image. The kernel ker f consists of all elements mapped to the identity (or zero in additive contexts). The image Im f is the set of all outputs. The first isomorphism theorem states that A/ker f is isomorphic to Im f; in particular, f is injective iff ker f is trivial, and surjective iff Im f equals the codomain.

Harvested insights from homomorphisms underpin much of algebra. They allow comparison of structures, construction of quotient objects, and passage to related objects via exact sequences in homological algebra. Variants appear across algebraic categories, including algebra, ring, and module homomorphisms, as well as the broader notion of morphisms in category theory.