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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

  • elements? I'm glad you asked, because yes, yes there are.

  • 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.

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  • Am I right?

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

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B2 commutative ring identity multiplication mod finite

Ring Examples (Abstract Algebra)

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    林宜悉 posted on 2020/03/06
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