## 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
*

* *

*Then, .*

**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,

and

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.

**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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