Quotient Rings (Pt. II)
Point of Post: This is a continuation of this post.
What shall now see though is that this property characterizes in a sense. In particular:
Theorem (Universal Characterization of Quotient Rings): Let be a ring and . Suppose that is a ring with a fixed epimorphism with . Moreover, suppose that is such that whenever there exists a morphism , for some ring , with there exists a unique morphism such that the following diagram commutes
Proof: Consider that since and we have that there exists a unique morphism with . Now, since and we know there exists a unique morphism such that . Thus,
Now, since is an epimorphism we may conclude from that and since is an epimorphism we may conclude from that . Thus, and are isomorphisms with . The conclusion follows.
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.