Isomorphism theorems

In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.

The isomorphism theorems were formulated in some generality for homomorphisms of modules by Emmy Noether in her paper , which was published in 1927 in Mathematische Annalen. Less general versions of these theorems can be found in work of Richard Dedekind and previous papers by Noether.

Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern

Three years later, B.L. van der Waerden published his influential Moderne Algebra the first abstract algebra textbook that took the groups-rings-fields approach to the subject. Van der Waerden credited lectures by Noether on group theory and Emil Artin on algebra, as well as a seminar conducted by Artin, Wilhelm Blaschke, Otto Schreier, and van der Waerden himself on ideals as the main references. The three isomorphism theorems, called homomorphism theorem, and two laws of isomorphism when applied to groups, appear explicitly.

Below we present four theorems, labelled A, B, C and D. They are often numbered as "First isomorphism theorem", "Second..." and so on; however, there is no universal agreement on the numbering. Here we give some examples of the group isomorphism theorems in the literature. Notice that these theorems have analogs for rings and modules.

It is less common to include the Theorem D, usually known as the lattice theorem or the correspondence theorem, to one of isomorphism theorems, but when they do, it is the last one.

Diagram for theorem B3. The two quotient groups (dotted) are isomorphic.

This theorem is sometimes called the isomorphism theorem,[10] diamond theorem[12] or the parallelogram theorem.[13]

The correspondence theorem (also known as the lattice theorem) is sometimes called the third or fourth isomorphism theorem.

The Zassenhaus lemma (also known as the butterfly lemma) is sometimes called the fourth isomorphism theorem.[14]

In the second isomorphism theorem, the product SN is the join of S and N in the lattice of subgroups of G, while the intersection S ∩ N is the meet.

The third isomorphism theorem is generalized by the nine lemma to abelian categories and more general maps between objects.

The statements of the theorems for rings are similar, with the notion of a normal subgroup replaced by the notion of an ideal.

Let R be a ring. Let S be a subring of R, and let I be an ideal of R. Then:

In the following, "module" will mean "R-module" for some fixed ring R.

Let M and N be modules, and let φ : M → N be a module homomorphism. Then:

To generalise this to universal algebra, normal subgroups need to be replaced by congruence relations.