Math 361, Spring 2014, Assignment 1

Carefully define the following terms, then give one example and one non-example of each:

1. Homomorphism (of rings).
2. Kernel (of a ring homomorphism).
3. Ideal.
4. Quotient ring.

Carefully state the following theorems (you need not prove them):

1. Fundamental theorem on ring homomorphisms.

Solve the following problems:

1. Section 26, problems 4, 12, 13, 14, 15, and 17.

--------------------End of assignment--------------------

Solutions:

Definitions:

1. Homomorphism (of rings).

   Definition: A map $\phi$ of a ring $R$ into a ring $R'$ is a homomorphism if it satisfies:

   $$\phi(a+b)=\phi(a)+\phi(b)$$ $$\phi(ab)=\phi(a)\phi(b)$$ for all $a,b\in R$

   Example:

   The function $\phi :\mathbb{Z}\rightarrow\mathbb{Z}_n$ defined by $\phi(z) = z\enspace mod\enspace n$ is a ring homomorphism

   Non-Example:

   There is no homomorphism from $\phi :\mathbb{Z}_n\rightarrow\mathbb{Z}$ when $n>1$.

2. Kernel (of a ring homomorphism).

   Let a map $\phi : R\rightarrow R$ be a homomorphism of rings. The subring

   $$\phi^{-1}[0'] = \{r\in R|\phi(r) = 0'\}$$

   is the kernel of $\phi$, denoted $Ker(\phi)$.

   Example:

   The function $\phi :\mathbb{Z}\rightarrow\mathbb{Z}_n$ defined by $\phi(z) = z\enspace mod\enspace n$ is a ring homomorphism with kernel $n\mathbb{Z}$

   Non-Example:

   Since there is no homomorphism from $\phi :\mathbb{Z}_n\rightarrow\mathbb{Z}$ when $n>1$, there is no kernel.

3. Ideal.
4. Quotient ring.