## Before the Lecture

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## 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).

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## In Class

In class:

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

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## 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.

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