## Fourth Ring Isomorphism Theorem (Lattice Isomorphism Theorem) (Pt. I)

**Point of Post: **In this post we discuss the notion of the fourth isomorphism theorem, but phrase it more in terms of lattices and Galois connections than is often done in most texts.

*Motivation*

** The fourth isomorphism theorem is much more involved than the other three isomorphism theorems (in content), but is perhaps one of the most useful results in basic ring theory. The fourth isomorphism theorem answers a fundamental question surely anyone trying to do actual ring theory (in the sense of dealing with real live rings, i.e. examples) will ask: “Is there a way I can find all the ideals of if I know all (or even better, just some of) the ideals of ?” Since quotient rings will often popup and ideals are of integral importance it makes sense that this is an important question. The beauty of the fourth isomorphism theorem is not only do we get a resounding “Yes!” to our question, but the connection between ideals turns out to be much, much more than we ever could have anticipated. Indeed, it turns out that there is a natural and astoundingly simple connection between subrings of containing and subrings of which besides being simple has many added properties. Moreover, we shall see that the map between these two sets of subrings will reduce nicely to a map between ideals of containing and ideals of .**

A nice way to picture precisely what the fourth ring isomorphism theorem says involves Hasse diagrams. Namely, what we shall see is that there is a natural correspondence between the lattice (we will show it is a lattice) of ideals of containing and the ideals of which is such that if the Hasse diagram for the first lattice is of the form

then the Hasse diagram for the second lattice will be

*Order Theoretic Prerequisites*

Let be a lattice. We call a *sublattice *of if is itself a lattice. We denote this by when convenient. It’s important to note that even if is a complete lattice and we need not have that is a complete lattice. An example of this can be seen in topology where one considers the topology of some space and note that while is a complete lattice and yet need not be closed under arbitrary infima (i.e. while is in the usual topology on the infimum of this collection of sets is which is not in the topology). Thus, we call a *complete sublattice *of it’s complete as a lattice. It’s obvious that such properties of lattices such as modularity are preserved under sublattices. The only thing we prove about sublattices is the following fact:

**Theorem: ***Let be a lattice. Define for some to be the set of all elements with . Then, is a sublattice of . Furthermore, if is complete then is a complete sublattice and for all .*

**Proof: **Let be arbitrary. Note then that since we have by definition that and evidently since we have that and so . Since were arbitrary the conclusion follows.

Suppose now that is complete. To show that is complete it evidently suffices to show that for every since then evidently each then has a supremum and . But, this is fairly evidently follows immediately from the same logic as the reason why . The conclusion follows.

Next, let and be posets. A mapping is called *monotone *if implies (i.e. that it preserves order). A *monotone Galois connection *between and is a pair of monotone maps and with the property that if and only if where and . In this context is called the *lower adjoint *and the *upper adjoint*.

Finally, let and two lattices (where and may be different operations) we call a map a *lattice homomorphism *if and for all . If is bijective we call a *lattice isomorphism *and it’s easy to see that is a lattice homomorphism . A c*omplete lattice homomorphism *is a map , where are complete lattices, which satisfies for every and a c*omplete lattice isomorphism *is a bijective complete lattice homomorphism–evidently then is a complete lattice isomorphism. It’s easy to see that every lattice or complete lattice homomorphism is monotone.

*The Fourth Isomorphism Theorem*

Before we actually state the fourth ring isomorphism theorem we make note of a quick fact about the lattice for some ring . Indeed, let and define to be the set of all ideals of containing . From the above theorems and comments we may conclude that is a modular complete sublattice of . With this in mind we see that:

**Theorem(Fourth Isomorphism Theorem/Lattice Isomorphism Theorem): ***Let be a ring and be an ideal of . The map *

*(where is the canonical projection) is a bijection which reduces to a complete lattice isomorphism*

*Moreover and form a montone Galois connection between and .*

**Proof: **We first show that is a bijection. This turns out to be relatively simple. Suppose that are subrings of are such that . Let then and so there exists some such that or that and thus and thus . The opposite inclusion follows similarly and thus we may conclude that is injective. To prove that is surjective we merely note that if is a subring of then is a subring of and since we may conclude that is a subring of containing , surjectivity then follows since (since is surjective).

What we next prove that . We first note that –this is immediate from the fact that for any we have that is an ideal of since is epic. The proof that the opposite inclusion is true we merely note that if then is an ideal of since we know the preimage of ideals under morphisms are ideals. Since we may thus conclude that and since (once again because is epic) surjectivity follows.

**References:**

1. Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

2. Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. *Basic Abstract Algebra*. Cambridge [Cambridgeshire: Cambridge UP, 1986. Print.

[…] Point of Post: This is a continuation of this post. […]

Pingback by Fourth Ring Isomorphism Theorem (Lattice Isomorphism Theorem) (Pt. II) « Abstract Nonsense | July 9, 2011 |

[…] know from the fourth ring isomorphism theorem that the ideals of are precisely with . But, by the third ring isomorphism theorem we know […]

Pingback by Prime Ideals (Pt. II) « Abstract Nonsense | August 17, 2011 |

[…] now we’re cooking with fire since the fourth isomorphism theorem tells us this is the case only when the ideal structure of looks […]

Pingback by Maximal Ideals (Pt. I) « Abstract Nonsense | August 31, 2011 |

[…] more useful is the fact that the fourth isomorphism theorem clearly implies […]

Pingback by Maximal Ideals (Pt. II) « Abstract Nonsense | September 6, 2011 |

[…] now that is prime, then is an integral domain. Now, if is an ideal in then we know by the lattice isomorphism theorem that for some . But, by assumption that is a PID we know that for some and it’s not hard […]

Pingback by PIDs (Pt. II) « Abstract Nonsense | October 22, 2011 |