Section 3.1 Part A: Rings

What methods are common among the following?

\mathbb{Z}, k\mathbb{Z} where k\ge 2, \mathbb{Z}_n, \mathbb{Q}, $\mathbb{R}$, \mathbb{C}

Answering this question leads to the definition of a ring.

Definition

A ring is a non empty set R with two operations, usually written as addition a+b, and multiplication ab, that satisfy the following axioms:

  1. If a,b\in R, then a+b\in R (Closure under Addition)
  2. For all a,b,c\in R, we have $a+(b+c)=(a+b)+c$
  3. For all a,b\in R we have a+b=b+a (Associative)
  4. There exists an element 0_R\in R such that for all a\in R we have a+0_R=a=0_R+a (Additive Identity Element)
  5. For each a\in R, the equation at a+x=0_R has a solution x\in R
  6. If a,b\in R, then ab\in R (Closure under Multiplication)
  7. For all a,b\in R, we have a(bc)=(ab)c (Associativity in Multiplication)
  8. For all a,b,c\in R, we have a(b+c)=ab+ac and (a+b)c=ac+bc

Note that commutativity isn’t explicitly written. That’s because multiplying on the left and right can be different. There some rings that are commutative. They are called commutative rings.

Definition:

A commutative ring is a ring R that satisfies

9. For all a,b\in R we have ab=ba (Multiplicative Identity)

All of these definitions may be a little confusing. But with a few examples will become easier to understand:

  1. \mathbb{R}^3 where it acts normal under addition but uses a cross product for multiplication:
    1. No. It isn’t associative. For example $latex \begin{align*}\hat{i}\times (\hat{i}\times \hat{k})&=(\hat{i}\times \hat{i})\times \hat{k}\\ \hat{i}\times (-\hat{j} ) & = 0 \\ -\hat{k} & \neq 0\end{align*}#
  2. M(\mathbb{R} ) ( 2×2 real value matrices) where they act as usual under addition but AB=B^T AB
    1. No. They are not associative. For example \begin{align*} (AB)C&= A (BC)\\ (B^T A B )C &= A (C^T BC)\\ C^T (B^T A B ) C & \neq (C^T B C )^T A (C^T B C)\end{align*}
  3. M(\mathbb{Z} acting as it normally does.
    1. Yes.
  4. M(\mathbb{Q} acting as it normally does.
    1. Yes.
  5. M(\mathbb{Z}_n acting as it normally does.
    1. Yes.

We didn’t explicitly mention commutitivity above, so it shall be defined here.

Definition:

An integral domain is the commutative ring R with identity 1_R\neq 0_R ( In other words R is a ring with at least two elements) such that:

11.  For a,b\in R if ab=0_R, then a=0_R or b=0_R (no zero divisor)

Here are a few examples that meet these conditions: \mathbb{Z}_p (p is prime), \mathbb{Z}\mathbb{Q}\mathbb{R}\mathbb{C}.

Definition:

A field is a commutative ring with an identity that satisfies:

12.  For each a\neq 0_R in R, the equation ax=b has a solution x in R.

So far only a few types of Rings have been mentioned. Below is a tree describing these various types:

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