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Math4302 Modern Algebra (Lecture 24)

Rings

Definition of ring

A ring is a set R with binary operation + and \cdot such that:

• (R,+) is an abelian group.
• Multiplication is associative: (a\cdot b)\cdot c=a\cdot (b\cdot c).
• Distribution property: a\cdot (b+c)=a\cdot b+a\cdot c, (b+c)\cdot a=b\cdot a+c\cdot a. (Note that \cdot may not be abelian, may not even be a group, therefore we need to distribute on both sides.)

Note

a\cdot b=ab will be used for the rest of the sections.

Examples of rings

(\mathbb{Z},+,*), (\mathbb{R},+,*) are rings.

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(2\mathbb{Z},+,\cdot) is a ring.

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(M_n(\mathbb{R}),+,\cdot) is a ring.

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(\mathbb{Z}_n,+,\cdot) is a ring, where a\cdot b=a*b\mod n.

e.g. in \mathbb{Z}_{12}, 4\cdot 8=8.

Tip

If (R+,\cdot) is a ring, then (R,\cdot) may not be necessarily a group.

Properties of rings

Let 0 denote the identity of addition of R. -a denote the additive inverse of a.

• 0\cdot a=a\cdot 0=0
• (-a)b=a(-b)=-(ab), \forall a,b\in R
• (-a)(-b)=ab, \forall a,b\in R

Proof

1. 0\cdot a=(0+0)\cdot a=0\cdot a+0\cdot a, by cancellation, 0\cdot a=0.
   Similarly, a\cdot 0=0\cdot a=0.

2. (a+(-a))\cdot b=0\cdot b=0 by (1), So a\cdot b +(-a)\cdot b=0, (-a)\cdot b=-(ab). Similarly, a\cdot (-b)=-(ab).

3. (-a)(-b)=(a(-b)) by (2), apply (2) again, -(-(ab))=ab.

Definition of commutative ring

A ring (R,+,\cdot) is commutative if a\cdot b=b\cdot a, \forall a,b\in R.

Example of non commutative ring

(M_n(\mathbb{R}),+,\cdot) is not commutative.

Definition of unity element

A ring R has unity element if there is an element 1\in R such that a\cdot 1=1\cdot a=a, \forall a\in R.

Note

Unity element is unique.

Suppose 1,1' are unity elements, then 1\cdot 1'=1'\cdot 1=1, 1=1'.

Example of field have no unity element

(2\mathbb{Z},+,\cdot) does not have unity element.

Definition of unit

Suppose R is a ring with unity element. An element a\in R is called a unit if there is b\in R such that a\cdot b=b\cdot a=1.

In this case b is called the inverse of a.

Tip

If a is a unit, then its inverse is unique. If b,b' are inverses of a, then b'=1b'=bab'=b1=b.

We use a^{-1} or \frac{1}{a} to represent the inverse of a.

Let R be a ring with unity, then 0 is not a unit. (identity of addition has no multiplicative inverse)

If 0b=b0=1, then \forall a\in R, a=1a=0a=0.

Definition of division ring

If every a\neq 0 in R has a multiplicative inverse (is a unit), then R is called a division ring.

Definition of field

A commutative division ring is called a field.

Example of field

(\mathbb{R},+,\cdot) is a field.

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(\mathbb{Z}_p,+,\cdot) is a field, where p is a prime number.

Lemma \mathbb{Z}_p is a field

\mathbb{Z}_p is a field if and only if p is prime.

Proof

If \mathbb{Z}_n is a field, then n is prime.

We proceed by contradiction. Suppose n is not a prime, then d|n for some 2\leq d\leq n-1, then [d] does not have inverse.

If [d][x]=[1], then dx\equiv 1\mod n, so dx-1=ny for some y\in \mathbb{Z}, but d|dx, and d|ny, so d|1 which is impossible.

Therefore, n is prime.

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If p is prime, then \mathbb{Z}_p is a field.

Since p is a prime, then \operatorname{gcd}(m,n)=1 for 1\leq m\leq n-1. So 1=mx+ny for some x,y\in \mathbb{Z}_p. Then [x] (the remainder of x when divided by p) is the multiplicative inverse of [m]. [m][x]=[mx]=[1-ny]=[1].