Please Login to access more options.
|
Sun |
Mon |
Tue |
Wed |
Thu |
Fri |
Sat |
Make sure you finish problems 5 and 6 from Friday. Then please read chapters 12 and 13 in the text. Before Monday, please make sure you submit a written solution to one of the problems from this week.
- In class I gave you a challenge. Find a ring $R$ and a subring $S$ so that that both $R$ and $S$ have a unity, but the unity in $S$ is not the unity in $R$.
- We also asked the following in class:If a ring has a unity, must that unity be unique? If a ring element has a multiplicative inverse, must that inverse be unique? Prove these, or disprove them.
If you want to master the material we are learning, I strongly suggest that each week you pick some additional problems from the text to help reinforce what we are doing.
- Chapter 12: 1,2,3,4,8,12,13,14, 18, 20. I'll let you read the others. If you see one that looks interesting, then do it. Challenge yourself.
- Chapter 13: You can do any of them, though the problems before 30 are related to what we did this week.
See you Monday.
In class Work
I'll be putting a few notes in here for in class Monday.
Work with $\mathbb{Z}_p[i]=\{a+bi \mid a,b\in \mathbb{Z}_p \}$ for several values of $p$. Any guess as to when is it a field? Look at $p=2,3,5$. For each, construct a multiplication table.
Show that $\mathbb{Z}[\sqrt{d}]$ is an integral domain. Show that $\mathbb{Q}[\sqrt{d}]$ is a field.
Show that $\phi:\mathbb{Z}\to\mathbb{Z}_n$ is a ring homomorphism.
Let $R$ be a ring with unity with characteristic $c$. Show that $\phi:\mathbb{Z}_c\to R$ is a ring homomorphism.
Suppose $R$ is a ring and $A$ is a subring of $R$. We know that $R/A$ is already an Abelian group. What would it take for $R/A$ to be a ring? What properties must we satisfy? What would we need to guarantee that $R/A$ is an integral domain? What would we need to guarantee that $R/A$ is a field?
Define nilpotent and idempotent. Find a nilpotent 2 by 2 nonzero matrix. Find a nonidentity, nonzero, matrix that is idempotent. If you are in an integral domain, what are the nilpotent and idempotent elements?
Under what conditions will $\mathbb{Z}_p[\sqrt{d}]$ be a field? Show that if $p=7$ and $d=3$, then this is a field.
When is it true that $(x+y)^n=x^n+y^n$?
If $a$ is nilpotent in a ring with unity, show that $1-a$ is a unit. If
For more problems, see AllProblems