Math 361, Spring 2022, Assignment 4

Read:

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

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

1. Homomorphism (of rings).
2. Unital homomorphism (of unital rings).
3. Pushforward (of a subring under a homomorphism; a.k.a. forward image).
4. Pullback (of a subring under a homomorphism; a.k.k. pre-image).
5. Image (of a ring homomorphism).
6. Kernel (of a ring homomorphism).
7. Ideal.
8. $R/I$ (the quotient of the ring $R$ by the two-sided ideal $I$).
9. Addition (in $R/I$, i.e. coset addition).
10. Multiplication (in $R/I$, i.e. coset multiplication).

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

1. Theorem concerning $\phi(0_R)$, where $\phi:R\rightarrow S$ is a ring homomorphism.
2. 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.
3. Theorem characterizing the properties of the pushforward of a subring (i.e. "The pushforward of a subring is a...").
4. Theorem characterizing the properties of the pullback of a subring (i.e. "The pullback of a subring is a...").
5. Theorem characterizing the special properties of kernels (i.e. "Kernels absorb..." or "Kernels are...").
6. Theorem characterizing ideals which contain units.
7. Theorem characterizing the ideals of a field.
8. Theorem characterizing when coset multiplication is well-defined (i.e. "Multiplication in $R/I$ is well-defined provided that $I$ is an...").
9. Equality test for elements of $R/I$.

Solve the following problems:

1. 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 $\mathrm{ker}(\phi)$).
2. (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.
3. Let $\pi:R\rightarrow R/I$ be the canonical projection defined above. Calculate $\mathrm{ker}(\pi)$.
4. Prove that $R/\{0\}$ is always isomorphic to $R$ itself. (Hint: use the your calculation of $\mathrm{ker}(\pi)$ from the last problem.)
5. Prove that $R/R$ is always a zero ring. (Hint: use the equality test for cosets.)
6. 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 $\mathrm{ker}(\phi)$.
7. Repeat the above exercise with $\mathbb{Z}_2\times\mathbb{Z}_4$ in place of $\mathbb{Z}_2\times\mathbb{Z}_3$.
8. 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--------------------

Questions:

Definitions:

1. $R/I$ (the quotient of the ring $R$ by the two-sided ideal $I$).

Theorems:

1. 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.
2. Theorem characterizing the properties of the pushforward of a subring (i.e. "The pushforward of a subring is a...").
3. Theorem characterizing the properties of the pullback of a subring (i.e. "The pullback of a subring is a...").

Definitions and Theorems:

https://drive.google.com/file/d/16HpHcJuiyCkTaeRBan3IsfSl1WO6qCY6/view?usp=sharing

Exam Preparation: https://drive.google.com/file/d/1YWB5VtKh7FsHVT8_SBlD0yQ2N4Ylmc1E/view?usp=sharing

Problems:

https://drive.google.com/file/d/15RrqY0HgCub1eHbvhjdIkyzLSdOWZTIi/view?usp=sharing