Math 361, Spring 2021, Assignment 4

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

1. $R/I$ (the quotient of the ring $R$ by the ideal $I$).
2. $\pi:R\rightarrow R/I$ (the canonical projection of $R$ onto $R/I$).
3. $\mathbb{C}$ (the complex number system, which we have defined as a certain quotient ring).
4. $i$ (the so-called "imaginary" number whose square is $-1$).

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

1. Theorem concerning the definition of coset multiplication (i.e. "Multiplication of cosets in $R/I$ is well-defined if $I$ is...").
2. Fundamental Theorem of Ring Homomorphisms.

Solve the following problems:

1. (Complex inversion) Let $a+bi$ be any element of $\mathbb{C}$ other than $0_{\mathbb{C}}$ (i.e. assume that $a$ and $b$ are not both zero). Prove that $a+bi$ is a unit, with inverse $\frac{a}{a^2+b^2}-\frac{b}{a^2+b^2}i$.
2. Suppose that $\phi:\mathbb{Z}\rightarrow\mathbb{Z}_2\times\mathbb{Z}_3$ is a ring homomorphism satisfying $\phi(1)=(1,1)$. Compute $\phi(2)$. (Hint: $\phi(2)=\phi(1+1)$.)
3. Let $\phi$ be as in the previous problem. Compute a table of values for $\phi$, showing enough lines to get a clear sense of the pattern in the outputs of $\phi$.
4. With $\phi$ as above, describe $\ker(\phi)$.
5. With $\phi$ as above, describe $\mathrm{im}(\phi)$.
6. Prove that $\mathbb{Z}_2\times\mathbb{Z}_3$ is isomorphic to $\mathbb{Z}_6$. (Hint: use the results of the previous problems, together with the Fundamental Theorem of Ring Homomorphisms.)
7. Repeat all of the above exercises except the last, this time supposing that $\phi$ is a ring homomorphism from $\mathbb{Z}$ to $\mathbb{Z}_2\times\mathbb{Z}_2$ satisfying $\phi(1)=(1,1)$.
8. We will see next week that the ring $\mathbb{Z}_2\times\mathbb{Z}_2$ is not isomorphic to $\mathbb{Z}_4$. Why does this not contradict the results of the previous exercises?

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