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
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.
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.
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.