Lecture 6

Navigation

• Back to Main Page.
• Back to the Lectures.
• Before the Lecture
• Related Problems
• In Class
• Outcomes

Before the Lecture

• Read Section 3.1:
  • Definition and basic properties of rings.
  • Skip Examples 3.10 and 3.11.
  • Subtraction and multiplication by integers.
  • Integral Domains.
  • Subrings.
  • Integers modulo $m$.
  • Units and divisbility.
• Watch the videos related to this section (after reading it):
  • General Remarks on rings.
  • Examples of rings.
  • Other operations (subtraction, multiplication by integers, powers).
  • Integers modulo \(m\).
  • Integral domains (and zero divisors).
  • Subrings (including Gaussian integers, \(\mathbb{Q}[\sqrt{2}]\)).
  • Units.
  • Problem 3.2.
  • Computations with Rings in Sage.
• Write down all questions about the above topics to bring to our (online) lecture. (You can also type them in the file "Questions.tex" in SageMathCloud.) Comments about the videos are welcome!
• Work on the assigned problems for these sections. (See Related Problems below.) You don't need to finish them, but try to work on as many as you can and the bring your questions to class.

 

Related Problems

The "turn in" problems are due on 06/19 by 11:59pm.

Section 3.1:	Turn in: 3.1(vi), 3.3(ii), 3.15(i).
	Extra Problems: 3.1 except (v) and (viii), 3.2, 3.3, 3.6, 3.8(i), 3.13, 3.15(i), (ii).

 

In Class

In class:

• We will discuss the reading and pace.
• I will discuss the main points.
• I will answer questions about the sections covered.
• I will answer any other questions.
• We can work on the HW problems.

 

Outcomes

After the assignment (reading and videos before class) and class, you should:

• know the definition, basic properties and examples of rings;
• be able to prove basic properties of abstract rings (Problems 3.2, 3.3);
• understand the difference between the multiplication of two elements of a ring and the multiplication of one element of a ring times an integer;
• know the definition and basic properties of an integral domain (Problems 3.8(i) and 3.15(i) and (ii));
• know the definition of a subring and know how to determine if a subset of a ring is a subring (Problem 3.13 and 3.15(i) and (ii));
• understand the rings $\mathbb{Q}[\sqrt{2}]$ (a subring of $\mathbb{R}$) and $\mathbb{Z}[\mathrm{i}]$ (a subring of $\mathbb{C}$);
• understand the ring of integers modulo $m$, denoted by $\mathbb{I}_m$ in the text, and know how to perform computations in this ring (Problem 3.6);
• know the definition of units and know how to find them (Problem 3.6);
• understand the generalization of divisibility on the context of abstract rings.