## Section 3.1 Rings Part B: Analysis

Cartesian products are very interesting. They essentially take everything that has done thus far and combines it.  An example given is a cartesian product where addition and multiplication are defined as The idea that this is valid for any two cartesian sets, given a few conditions, is fascinating. Yet I wonder how this would be …

## Section 3.1 Part A: Rings

What methods are common among the following? , where , , , \$\mathbb{R}\$, Answering this question leads to the definition of a ring.

## Section 3.1 Rings Part A: Analysis

In this section Hungerford defines a ring and gives several different examples. From them I gather that there are many different types of rings. But I think the type of rings that will be the most interesting shall be those where every axiom is met (see page 44). When looking at a ring outside of …

## Section 2.3: The Arithmetic of \$latex\mathbb{Z}_p\$ (p Prime) and \$latex\mathbb{Z}_n\$\$

In my analysis of this section I noted that Hungerford often switches between the conventional modular arithmetic notation (with the brackets []) and then uses the number by itself. While both are completely valid certain uses are better than others. That is the notation is much better than a if there are integers in being used …

## Section 2.3 Analysis

Section 2.3: The Structure of (p Prime) and We’ve mentioned that Modular Arithmetic is used in cryptography. However after studying this section it seems more likely that (p being a prime number) to be used in cryptography since there rules governing, for example, units are even more strict.

## Section 2.2: Modular Arithmetic

Given any set , where we define two operations: and . and

## Section 2.2: Modular Arithmetic Analysis

Modular Arithmetic   Analysis: It is clear that there are an n number of unique sets of Z_n but is there a way to calculate [a]*[b] without counting the final value of a*b and finding where that lies mod n? It’s very interesting that everywhere we look we can always find some sort of pattern. …

## Hungerford’s Section 2.1

Hungerford’s Section 2.1 Analysis Thus far in Hungerford’s book I have yet to encounter something super difficult to understand. Proof techniques, of course, shall return to my memory as I study and practice but the material, especially in Section 2.1 is very straight forward. The most difficult part I have is trying to see how …

## Section 2.1: Congruence and Congruence Classes

Congruence The concept of congruence initially may seem a bit confusing. I hope to explain the definition in a way that may be easier to understand. Definition of Congruence:

## Prime Numbers

Primes As mentioned before: an integer is prime if the only divisors of said integer are ±1, or ± said-integer, but 0 and ±1 are not considered to be prime. (Think about it any integer divides