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• The key structures in Abstract Algebra are groups, rings, fields, vector spaces,

• and modules. You start with groups because the other four structures are

• built upon them. All of these concepts are fairly abstract, so it's helpful to

• learn lots of concrete examples to help keep you grounded and test out

• everything that you learn. Today, let's look at examples of rings.

• Before we dive into the examples, let's remind ourselves of the definition of a ring.

• Loosely speaking, a ring is a set of elements where you can freely add, subtract, and

• multiply with the usual rules of arithmetic. And while addition is

• commutative, multiplication may not be

• Technically speaking, a ring R is a set

• of elements with two operations: addition and multiplication.

• It's an abelian group under addition. Multiplication obeys the associative

• property, and addition and multiplication are connected by the left and right

• distributive laws. In a ring, multiplication may not be commutative. If

• it is, we call R a commutative ring.

• And if the ring has a multiplicative identity

• one, we call it a ring with identity. Every ring has the additive identity 0,

• since it's a group under addition. So when you say ring with identity,

• everyone knows that you mean ring with multiplicative identity.

• A classic example of a ring are the integers.

• The set of integers is denoted with a double stroke Z.

• This is from the German word Zahlen, meaning number. Historically,

• number meant integer. That's why number theory is a study of the integers.

• Nowadays, however, number has a more general meaning, but we stick with the

• letter Z for the integers. Another example of a ring is a set of

• polynomials with real coefficients, you can freely add, subtract, and multiply any

• two polynomials, and you'll get a third polynomial. Each polynomial has an

• additive inverse. For any polynomial F, its additive inverse is a polynomial

• where you take the opposite of each term. The additive identity is zero.

• Don't forget that zero is a polynomial. On the multiplication side, there is an

• identity element: the number one. And for real polynomials, multiplication is

• commutative. This is a commutative ring with an identity element.

• In this last example, we looked at polynomials with real number coefficients.

• But you can use any ring for the coefficients: the integers, the complex numbers, or how

• about the integers mod n? You can even have polynomials where the coefficients

• are matrices. Any ring will do. So if you are handed a ring R, you can use it

• to make a new ring: the ring of polynomials with coefficients in R.

• Here is its notation. Be careful you use brackets and not parentheses. They mean

• different things. Brackets means the set of polynomials;

• while parentheses means the set of rational functions, which are fractions of polynomials.

• So far every ring that we've seen have been commutative, and

• has had an identity element for multiplication. Let's now look at rings

• where this is not the case. An example of a ring without an identity element is the

• set of even integers. If you add, subtract, or multiply any two even numbers, you'll

• get another even number. But the number one is not in this set, because one is

• odd. So while this is a commutative ring, it does not have an identity element for multiplication.

• Let's now look at a ring that's not commutative. The classic

• example here are matrices. For instance, the set of 2 by 2 matrices with whole

• number entries are a ring, but it's not commutative. To see why, consider the two

• matrices 1 2 3 4 and 1 0 negative 1 1. Here, A times B does not equal B times A.

• So this ring is not commutative. We call this a non-commutative ring. As a

• consolation prize, however, it does have an identity element: the 2 by 2 identity matrix.

• Just for fun, let's see a ring that is not commutative and does not

• have an identity: the 2 by 2 matrices with even entries. This definitely does

• not have an identity, since the identity matrix has ones along the diagonal and

• one is an odd number. This poor ring is having an identity crisis.

• Already we've seen a wide variety of rings: commutative, non-commutative,

• identity, no identity - but all the examples have had something in common:

• they have all been infinite. Are there rings with only a finite number of

• The integers mod n are a finite ring with only n elements.

• Here is the notation for this ring .

• You may recognize this from your study of groups. This is the notation for a quotient group.

• Z is a group and n Z is a normal subgroup.

• The quotient group is the group of cosets. This notation is carried over to rings.

• Here, Z is the ring, and n Z is an ideal. In this context, we call it a quotient ring.

• The integers mod n is a finite ring. It is also commutative, and contains an

• identity element 1. This gives us a nice list of finite rings: the integers mod 2,

• mod 3, mod 4, mod 5, and so on. But something special happens when N is a

• prime number. If n is prime, the integers mod n is now a field. Here's where you

• need to be careful. Every field is a ring, but not every ring is a field.

• If you were to draw a Venn diagram, the set of all fields would lie inside the set the

• all rings. And since every ring is a commutative group under addition, the set

• of rings lies inside the set of groups. From the examples that we've seen,

• we can add more details to this diagram. Inside the set of rings are the

• commutative rings and the rings with identity. Every field is commutative and

• has an identity, so the set of fields lies in the overlap between commutative rings

• and rings with identity. I know what you're thinking you've shown

• me lots of examples of rings: infinite rings, finite rings, commutative rings,

• non-commutative rings... but you haven't yet shown me a finite non-commutative ring.

• This I'm going to leave as a challenge for you. Here's a hint: you've seen a

• finite ring: the integers mod n; a non commutative ring: the two by two matrices.

• In the comments below, help each other understand how you can combine these to

• make a finite non-commutative ring. By doing so, you'll be joining the

• Fellowship of the Rings.

• is zero. If you liked this video and would like to help us grow, please share

• it with someone, and if you know any captains of industry let them know that

• Socratica would be grateful for their support on Patreon! I mean who needs a yacht...

• Am I right?

The key structures in Abstract Algebra are groups, rings, fields, vector spaces,

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Ring Examples (Abstract Algebra)

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林宜悉 posted on 2020/03/06
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