Math 361, Spring 2022, Assignment 4

Read:[edit]

Section 26. (Note that we are skipping forward somewhat in the text; however we will shortly return to the earlier material in Section 20.)

Theorem concerning $\phi(0_R)$, where $\phi:R\rightarrow S$ is a ring homomorphism.
Examples of ring homomorphisms $\phi:R\rightarrow S$ to show that $\phi(1_R)$ may or may not equal $1_S$, even when $R$ and $S$ are both unital.
Theorem characterizing the properties of the pushforward of a subring (i.e. "The pushforward of a subring is a...").
Theorem characterizing the properties of the pullback of a subring (i.e. "The pullback of a subring is a...").
Theorem characterizing the special properties of kernels (i.e. "Kernels absorb..." or "Kernels are...").
Theorem characterizing ideals which contain units.
Theorem characterizing the ideals of a field.
Theorem characterizing when coset multiplication is well-defined (i.e. "Multiplication in $R/I$ is well-defined provided that $I$ is an...").
Equality test for elements of $R/I$.

Section 26, problems 4, 17, and 18 (hint: if $\phi:F\rightarrow S$ is a homomorphism defined on a field $F$, then there are not many possibilities for $\mathrm{ker}(\phi)$).
(Canonical projection) Suppose $I$ is an ideal of a ring $R$. Define a map $\pi:R\rightarrow R/I$ by the formula $\pi(r)=r+I$. Show that $\pi$ is an epimorphism, and that it is a unital epimorphism whenever $R$ is a unital ring.
Let $\pi:R\rightarrow R/I$ be the canonical projection defined above. Calculate $\mathrm{ker}(\pi)$.
Prove that $R/\{0\}$ is always isomorphic to $R$ itself. (Hint: use the your calculation of $\mathrm{ker}(\pi)$ from the last problem.)
Prove that $R/R$ is always a zero ring. (Hint: use the equality test for cosets.)
We shall see next week that there is one and only one ring homomorphism $\phi:\mathbb{Z}\rightarrow\mathbb{Z}_2\times\mathbb{Z}_3$ for which $\phi(1)=(1,1)$. Write the table of values for this homomorphism, then describe $\mathrm{im}(\phi)$ and $\mathrm{ker}(\phi)$.
Repeat the above exercise with $\mathbb{Z}_2\times\mathbb{Z}_4$ in place of $\mathbb{Z}_2\times\mathbb{Z}_3$.
By comparing the previous two exercises, see whether you can make any conjecture about the relationship between $\mathbb{Z}_{a}\times\mathbb{Z}_b$ and $\mathbb{Z}_{ab}$. (If you manage this then you will have re-discovered the ancient and beautiful Chinese Remainder Theorem (CRT), which we will study next week.)

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

Definitions:

Theorems:

Examples of ring homomorphisms $\phi:R\rightarrow S$ to show that $\phi(1_R)$ may or may not equal $1_S$, even when $R$ and $S$ are both unital.
Theorem characterizing the properties of the pushforward of a subring (i.e. "The pushforward of a subring is a...").
Theorem characterizing the properties of the pullback of a subring (i.e. "The pullback of a subring is a...").