Math 361, Spring 2021, Assignment 4
From cartan.math.umb.edu
Carefully define the following terms, then give one example and one non-example of each:[edit]
- $R/I$ (the quotient of the ring $R$ by the ideal $I$).
- $\pi:R\rightarrow R/I$ (the canonical projection of $R$ onto $R/I$).
- $\mathbb{C}$ (the complex number system, which we have defined as a certain quotient ring).
- $i$ (the so-called "imaginary" number whose square is $-1$).
Carefully state the following theorems (you do not need to prove them):[edit]
- Theorem concerning the definition of coset multiplication (i.e. "Multiplication of cosets in $R/I$ is well-defined if $I$ is..."): $\bigstar$ Theorem concerning the definition of coset multiplication (i.e. "Multiplication of cosets in $R/I$ is well-defined if $I$ is..."): When $I$ is an ideal, Multiplication of cosets in $R/I$ is well-defined.
- Fundamental Theorem of Ring Homomorphisms: Let $\phi : G \rightarrow G'$ be a group homomorphism with kernel $H$. Then $\phi[G]$ is a group, and $\mu : G/H \rightarrow \phi[G]$ given by $\mu(gH) = \phi(g)$ is an isomorphism. If $\gamma : G \rightarrow G/H $is the homomorphism given by $\gamma(g) = gH,$ then $\phi(g) = \mu\gamma(g)$ for each $g \in G$.
Solve the following problems:[edit]
- (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 $\frac{a}{a^2+b^2}-\frac{b}{a^2+b^2}i$.
- 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: $\phi(2)=\phi(1+1)$.)
- 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 $\phi$.
- With $\phi$ as above, describe $\ker(\phi)$.
- With $\phi$ as above, describe $\mathrm{im}(\phi)$.
- 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.)
- 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 $\phi(1)=(1,1)$.
- 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?
