4.2 Rings and Fields

Definitions

We are going to define a ring (R+⋅) or simply R.

Data:

• A set R of elements.

• An operation +:R×R→R called addition.

• An operation ⋅:R×R→R called multiplication.

Axioms:

• (R,+)is an abelian group.

• Multiplication is left and right distributive over addition:

• a*(b+c)=a*b+a*c and(a+b)*c=a*c+b*c for all a*b*c∈R

• Multiplication is associative: (a*b)*c=a(b*c) for all a*b*c∈R.

• Multiplication has a neutral element denoted by 1.

In this case, we say that (R,+,⋅) is an associative ring with an identity. We will use the term ring to denote an associative ring with an identity. As before, we usually say that R is a ring assuming that the operations in use are clear. The neutral element with respect to addition is denoted by 0 and is called zero. If a∈R is an arbitrary element, its inverse with respect to addition is denoted by -a. For any a,b∈R, the expression a+(-b) will be denoted by a-b for short. If in addition we have

• Multiplication is commutative: a*b=b*a for all a*b∈R.

The ring is said to be commutative.

Field

A field is a set F which is closed under two operations + and × such that:

• Fis an abelian group under+

• F-{0} the set F without the additive identity0is an abelian group under×

Alternative Definition: A field is a commutative ring where every non-zero element is invertible with respect to multiplication, and 1≠0. More explicitly, if we add the following conditions to a commutative ring:

• Every non-zero element is invertible with respect to multiplication: for every a∈R∖{0}, there exists an element b∈R such that a*b=b*a=1.

• 1≠0

The ring is said to be a field. In this case, the inverse element for a is denoted by a(-1).

Subring

Let R be a ring. We are going to define a subring T⊆R.

Data:

• A subset T⊆R.

Axioms:

• (T,+)⊆(R,+) is a subgroup.

• T is closed under multiplication.

• T contains1

Ideals

Suppose that (R+⋅) is a ring. I am going to define an ideal I in the ring R.

Data:

• A subset I⊆R.

Axioms:

• (I,+)⊆(R,+)is a subgroup.

• For any r∈R we have

  r*I={r*x|x∈I}⊆I and I*r={x*r|x∈I}⊆I

More Cheat Sheets ->