Table of contents |
2 Rings and Modules 3 General |
First we state the isomorphism theorems for groups, where they take a simpler form and state important properties of factor groups (also called quotient groups).
The isomorphism theorems are also valid for modules over a fixed ring R (and therefore also for vector spaces over a fixed field). One has to replace the term "group" by "R-module", "subgroup" and "normal subgroup" by "submodule", and "factor group" by "factor module".
The isomorphism theorems are also valid for rings, ring homomorphisms and idealss. One has to replace the term "group" by "ring", "subgroup" and "normal subgroup" by "ideal", and "factor group" by "factor ring".
The notation for the join in both these cases is "S + N" instead of "SN".
To generalise this to universal algebra, normal subgroups need to be replaced with congruencess.