isomorphism theorems on algebraic systems

In this entry, all algebraic systems are of the same type; they are all O-algebrasMathworldPlanetmathPlanetmath. We list the generalizationsPlanetmathPlanetmath of three famous isomorphism theorems, familiar to those who have studied abstract algebra in college.

Theorem 1.

If f:AB is a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from algebras A and B. Then

Theorem 2.

If BA are algebras and C is a congruencePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ( on A, then


where CB is the congruence restricted to B, and BC is the extensionPlanetmathPlanetmathPlanetmath of B by C.

Theorem 3.

If A is an algebra and CD are congruences on A. Then

  1. 1.

    there is a unique homomorphism f:A/A/𝔇 such that


    where the downward pointing arrows are the natural projectionsMathworldPlanetmath of A onto the quotient algebras (induced by the respective congruences).

  2. 2.

    Furthermore, if ker(f)=𝔇/, then

    • 𝔇/ is a congruence on A/, and

    • there is a unique isomorphismMathworldPlanetmathPlanetmath f:A/(A/)/(𝔇/) satisfying the equation f=[]𝔇/f. In other words,

Title isomorphism theorems on algebraic systems
Canonical name IsomorphismTheoremsOnAlgebraicSystems
Date of creation 2013-03-22 16:45:28
Last modified on 2013-03-22 16:45:28
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Theorem
Classification msc 08A05