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

**Point of Post: **This is a continuation of this post.

To prove that is a complete lattice isomorphism we begin by letting be arbitrary. We recall that is equal to . But, it’s trivial to see that since is surjective that we have

Now, let . We claim that

The inclusion

we get for free as a common set-theoretic fact. To see the reverse inclusion let be in . Fix some , since clearly we have that for some . I claim that for all from where the desired inclusion follows. To see this let be arbitrary, since there exists some for which . Thus, since is a morphism we have that and so . Since was arbitrary we have that is in . Thus, . Since was arbitrary the equality follows.

Putting these together with the previous result that is a bijection shows that is a complete lattice isomorphism as claimed.

It remains to show that and form a monotone Galois connection between and . For this theorem that particularly means that if and only if . But, this follows immediately from the fact that is monotone (since is a complete lattice isomorphism) and the obvious fact that the inverse of a monotone function is monotone.

Putting this all together the theorem follows.

*Remark: *It’s also fairly clear to see that is a semiring homomorphism. Moreover, if I would have defined the lattice of subrings we would see that the above could be extended to that case in the same manner.

*A Generalization*

Although not used quite as often the above lattice isomorphism theorem enables us to prove the following ‘generalization’:

*Generalized Fourth Isomorphism Theorem: **Let and be rings and an epimorphism. Then the map given by is a lattice ismorphism. Moreover, and form a monotone Galois connection between and .*

**Proof: **This follows immediately from the fourth ring isomorphism theorem and the first ring isomorphism theorem since the map is an isomorphism.

**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.

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