### Noether isomorphism theorem

In mathematics, specifically abstract algebra, the **isomorphism theorems** are three 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.

## Contents

## History

The isomorphism theorems were formulated in some generality for homomorphisms of modules by Emmy Noether in her paper *Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern* 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.

Three years later, B.L. van der Waerden published his influential *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.

## Groups

We first state the three isomorphism theorems in the context of groups. Note that some sources switch the numbering of the second and third theorems.^{[1]} Sometimes the lattice theorem is referred to as the *fourth isomorphism* theorem or the *correspondence theorem*.

### Statement of the theorems

#### First isomorphism theorem

Let *G* and *H* be groups, and let *φ*: *G* → *H* be a homomorphism. Then:

- The kernel of
*φ*is a normal subgroup of*G*, - The image of
*φ*is a subgroup of*H*, and - The image of
*φ*is isomorphic to the quotient group*G*/ ker(*φ*).

In particular, if *φ* is surjective then *H* is isomorphic to *G* / ker(*φ*).

#### Second isomorphism theorem

Let *G* be a group. Let *S* be a subgroup of *G*, and let *N* be a normal subgroup of *G*. Then:

- The product
*SN*is a subgroup of*G*, - The intersection
*S*∩*N*is a normal subgroup of*S*, and - The quotient groups (
*SN*) /*N*and*S*/ (*S*∩*N*) are isomorphic.

Technically, it is not necessary for *N* to be a normal subgroup, as long as *S* is a subgroup of the normalizer of *N*. In this case, the intersection *S* ∩ *N* is not a normal subgroup of *G*, but it is still a normal subgroup of *S*.

#### Third isomorphism theorem

Let *G* be a group. Let *N* and *K* be normal subgroups of *G*, with

*K*⊆*N*⊆*G*.

Then

- The quotient
*N*/*K*is a normal subgroup of the quotient*G*/*K*, and - The quotient group (
*G*/*K*) / (*N*/*K*) is isomorphic to*G*/*N*.

### Discussion

The first isomorphism theorem follows from the category theoretical fact that the category of groups is (normal epi, mono)-factorizable; in other words, the normal epimorphisms and the monomorphisms form a factorization system for the category. This is captured in the commutative diagram in the margin, which shows the objects and morphisms whose existence can be deduced from the morphism *f*: *G*→*H*. The diagram shows that every morphism in the category of groups has a kernel in the category theoretical sense; the arbitrary morphism *f* factors into $\backslash iota\; \backslash circ\; \backslash pi$, where *ι* is a monomorphism and *π* is an epimorphism (in a conormal category, all epimorphisms are normal). This is represented in the diagram by an object $\backslash ker\backslash ,\; f$ and a monomorphism $\backslash kappa:\; \backslash ker\backslash ,\; f\; \backslash rightarrow\; G$ (kernels are always monomorphisms), which complete the short exact sequence running from the lower left to the upper right of the diagram. The use of the exact sequence convention saves us from having to draw the zero morphisms from $\backslash ker\backslash ,\; f$ to *H* and $G\; /\; \backslash ker\backslash ,\; f$.

If the sequence is right split (i. e., there is a morphism *σ* that maps $G\; /\; \backslash ker\backslash ,\; f$ to a *π*-preimage of itself), then *G* is the semidirect product of the normal subgroup $\backslash operatorname\{im\}\backslash ,\; \backslash kappa$ and the subgroup $\backslash operatorname\{im\}\backslash ,\; \backslash sigma$. If it is left split (i. e., there exists some $\backslash rho:\; G\; \backslash rightarrow\; \backslash ker\backslash ,\; f$ such that $\backslash rho\; \backslash circ\; \backslash kappa\; =\; \backslash operatorname\{id\}\_\{\backslash ker\backslash ,\; f\}$), then it must also be right split, and $\backslash operatorname\{im\}\backslash ,\; \backslash kappa\; \backslash times\; \backslash operatorname\{im\}\backslash ,\; \backslash sigma$ is a direct product decomposition of *G*. In general, the existence of a right split does not imply the existence of a left split; but in an abelian category (such as the abelian groups), left splits and right splits are equivalent by the splitting lemma, and a right split is sufficient to produce a direct sum decomposition $\backslash operatorname\{im\}\backslash ,\; \backslash kappa\; \backslash oplus\; \backslash operatorname\{im\}\backslash ,\; \backslash sigma$. In an abelian category, all monomorphisms are also normal, and the diagram may be extended by a second short exact sequence $0\; \backslash rightarrow\; G\; /\; \backslash ker\backslash ,\; f\; \backslash rightarrow\; H\; \backslash rightarrow\; \backslash operatorname\{coker\}\backslash ,\; f\; \backslash rightarrow\; 0$.

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. It is sometimes informally called the "freshman theorem", because "even a freshman could figure it out: just cancel out the *K*s!"

## Rings

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

### First isomorphism theorem

Let *R* and *S* be rings, and let *φ*: *R* → *S* be a ring homomorphism. Then:

- The kernel of
*φ*is an ideal of*R*, - The image of
*φ*is a subring of*S*, and - The image of
*φ*is isomorphic to the quotient ring*R*/ ker(*φ*).

In particular, if *φ* is surjective then *S* is isomorphic to *R* / ker(*φ*).

### Second isomorphism theorem

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

- The sum
*S*+*I*= {*s*+*i*|*s*∈*S*,*i*∈*I*} is a subring of*R*, - The intersection
*S*∩*I*is an ideal of*S*, and - The quotient rings (
*S*+*I*) /*I*and*S*/ (*S*∩*I*) are isomorphic.

### Third isomorphism theorem

Let *R* be a ring. Let *A* and *B* be ideals of *R*, with

*B*⊆*A*⊆*R*.

Then

- The set
*A*/*B*is an ideal of the quotient*R*/*B*, and - The quotient ring (
*R*/*B*) / (*A*/*B*) is isomorphic to*R*/*A*.

## Modules

The statements of the isomorphism theorems for modules are particularly simple, since it is possible to form a quotient module from any submodule. The isomorphism theorems for vector spaces and abelian groups are special cases of these. For vector spaces, all of these theorems follow from the rank-nullity theorem.

For all of the following theorems, the word “module” will mean “*R*-module”, where *R* is some fixed ring.

### First isomorphism theorem

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

- The kernel of
*φ*is a submodule of*M*, - The image of
*φ*is a submodule of*N*, and - The image of
*φ*is isomorphic to the quotient module*M*/ ker(*φ*).

In particular, if *φ* is surjective then *N* is isomorphic to *M* / ker(*φ*).

### Second isomorphism theorem

Let *M* be a module, and let *S* and *T* be submodules of *M*. Then:

- The sum
*S*+*T*= {*s*+*t*|*s*∈*S*,*t*∈*T*} is a submodule of*M*, - The intersection
*S*∩*T*is a submodule of*S*, and - The quotient modules (
*S*+*T*) /*T*and*S*/ (*S*∩*T*) are isomorphic.

### Third isomorphism theorem

Let *M* be a module. Let *S* and *T* be submodules of *M*, with

*T*⊆*S*⊆*M*.

Then

- The quotient
*S*/*T*is a submodule of the quotient*M*/*T*, and - The quotient (
*M*/*T*) / (*S*/*T*) is isomorphic to*M*/*S*.

## General

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

A **congruence** on an algebra $A$ is an equivalence relation $\backslash Phi$ which is a subalgebra of $A\; \backslash times\; A$ endowed with the component-wise operation structure. One can make the set of equivalence classes $A/\backslash Phi$ into an algebra of the same type by defining the operations via representatives; this will be well-defined since $\backslash Phi$ is a subalgebra of $A\; \backslash times\; A$.

### First Isomorphism Theorem

Let $f:A\; \backslash rightarrow\; B$ be an algebra homomorphism. Then the image of $f$ is a subalgebra of $B$, the relation given by $\backslash Phi:f(x)=f(y)$ is a congruence on $A$, and the algebras $\backslash \; A/\backslash Phi\; \backslash $ and $\backslash text\{im\}\backslash \; f$ are isomorphic.

### Second Isomorphism Theorem

Given an algebra $A$, a subalgebra $B$ of $A$, and a congruence $\backslash Phi$ on $A$, let $\backslash Phi\_B\; =\; \backslash Phi\; \backslash cap\; (B\; \backslash times\; B)$ be the trace of $\backslash Phi$ in $B$ and $[B]^\backslash Phi=\backslash \{K\; \backslash in\; A/\backslash Phi:\; K\; \backslash cap\; B\; \backslash neq\backslash emptyset\backslash \}$ the collection of equivalence classes that intersect $B$.

Then (i) $\backslash Phi\_B$ is a congruence on $B$, (ii) $\backslash \; [B]^\backslash Phi$ is a subalgebra of $A/\backslash Phi$, and (iii) the algebra $[B]^\backslash Phi$ is isomorphic to the algebra $B/\backslash Phi\_B$.

### Third Isomorphism Theorem

Let $A$ be an algebra and $\backslash Phi,\; \backslash Psi$ two congruence relations on $A$ such that $\backslash Psi\; \backslash subseteq\; \backslash Phi$. Then $\backslash Phi/\backslash Psi=\; \backslash \{\; ([a\text{'}]\_\backslash Psi,[a$*]_\Psi): (a',a*)\in \Phi\} = [\ ]_\Psi \circ \Phi \circ [\ ]_\Psi^{-1} is a congruence on $A/\backslash Psi$, and $A/\backslash Phi$ is isomorphic to $(A/\backslash Psi)/(\backslash Phi/\backslash Psi)$.

## See also

- Butterfly lemma, sometimes called the fourth isomorphism theorem
- Lattice theorem, sometimes called the fourth isomorphism theorem
- Splitting lemma, which refines the first isomorphism theorem for split sequences

## Notes

## References

- Emmy Noether,
*Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern*, Mathematische Annalen**96**(1927) p. 26-61 - Colin McLarty, 'Emmy Noether’s ‘Set Theoretic’ Topology: From Dedekind to the rise of functors' in
*The Architecture of Modern Mathematics: Essays in history and philosophy*(edited by Jeremy Gray and José Ferreirós), Oxford University Press (2006) p. 211–35. -

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