A #~{ring} is a set ~R with two internal binary operations, addition ( + ) and multiplication ( # ), such that:

- ( ~R , + ) is an Abelian group, i.e. , &forall. ~x, ~y, ~z &in. ~R :
- ( ~x + ~y ) + ~z _ = _ ~x + ( ~y + ~z ) ,
- &exist. 0 &in. ~R such that _ 0 + ~x = ~x . _ [ 0 is called the #~{zero element} ]
- &exist. -~x &in. ~R _ such that _ ~x + -~x = 0 ,
- ~x + ~y _ = _ ~y + ~x .

- ( ~R , # ) is a semigroup, i.e. associative : _ ~x ( ~y ~z ) _ = _ ( ~x ~y ) ~z.

_ - The operations are #~{distributive}, i.e. , &forall. ~x, ~y, ~z &in. ~R :
- ~x ( ~y + ~z ) _ = _ ~x ~y + ~x ~z ,
- ( ~x + ~y ) ~z _ = _ ~x ~z + ~y ~z .

Lemma: _ 0 ~x _ = _ 0 , _ &forall. ~x &in. ~R, _ [ 0 ~x = ( 0 + 0 ) ~x = 0 ~x + 0 ~x _ => _ 0 ~x = 0 , _ by ]

If &exist. element _ 1 &in. ~R _ such that _ 1 ~x = ~x 1 = ~x , _ &forall. ~x &in. ~R, then this is called the #~{identitiy element}, and ~R is said to be a #~{ring with identity}.

A ring, ~R, in which multiplication is commutative, i.e. _ ~x ~y _ = _ ~y ~x , _ &forall. ~x, ~y &in. ~R , _ is called a #~{commutative ring}.

The set of integers (positive, negative and zero), _ #Z , _ under the normal operations of addition and multiplication.

The set _ \{ 0, 1, 2, ... , ~n - 1 \} _ under the operations of addition (mod ~n) and multiplication (mod ~n).

Consider the set \{ 0, 1, 2, ... , 5 \} _ with addition and multiplication (mod 6). _ Then _ 2 # 3 _ = _ 6 _ = _ 0 (mod 6) . _ So we have a ring in which two non-zero elements have a product of zero. We will return to this in the topic of integral domains, below.

array{ # ,#0,#1,#2,#3,#4,#5/#0,0,0,0,0,0,0/#1,0,1,2,3,4,5/ #2,0,2,4,0,2,4/#3,0,3,0,3,0,3/#4,0,4,2,0,4,2/#5,0,5,4,3,2,1} | #{Multiplication table mod 6} This is a commutative ring. Note that several elements do not have a multiplicative inverse, e.g. there is no ~x such that ~x # 4 = 1. So the set is not a group under multiplication. |

If instead we choose ~n to be a prime, then there are no elements whose product is zero.

array{ # ,#0,#1,#2,#3,#4,/#0,0,0,0,0,0/#1,0,1,2,3,4/ #2,0,2,4,1,3/#3,0,3,1,4,2/#4,0,4,3,2,1} | #{Multiplication table mod 5} This is a commutative ring in which every non-zero element has a multiplicative inverse, e.g. 1 # 1 , 2 # 3 , and 4 # 4. The set excluding zero is a group under multiplication, and so is a field as we shall see below. |

A subset _ ~S &subseteq. ~R _ is a #~{subring} of ~R if ~S is a ring in its own right, i.e.:

- ~S is a group under addition (forcibly abelian)
- ~S is closed under multiplication.

The distributive and associative rules will be "inherited" from ~R.

A subring, ~I , is a #~{left ideal} of ~R if _ ~x ~r &in. ~I , _ &forall. ~x &in. ~I , ~r &in. ~R. _ [ #~{right ideal} if _ ~r ~x &in. ~I ].

If ~R is a commutative ring then the left and right ideals of ~R coincide, so we just talk of an #~{ideal} of ~R.

If ~R is a commutative ring and ~I is an ideal of ~R, then under addition ~I is a subgroup of abelian group ~R, _ ( ~I , + ) _ < _ ( ~R , + ) , _ so we can form the quotient _ ~R ./ ~I , _ where addition is defined: _ ( ~r_1 + ~I ) + ( ~r_2 + ~I ) _ = _ ( ~r_1 + ~r_2 ) + ~I.

We now define multiplication on the quotient: _ ( ~r_1 + ~I ) ( ~r_2 + ~I ) _ = _ ( ~r_1 ~r_2 ) + ~I.

Multiplication is well defined, in the sense that if _ ~r_1' ~ ~r_1 _ ( i.e. _ ~r_1' = ~r_1 + ~x , _ some ~x &in. ~I ) _ then

( ~r_1' + ~I ) ( ~r_2 + ~I ) _ = _ ( ~r_1' ~r_2 + ~I ) _ = _ ( ( ~r_1 + ~x ) ~r_2 + ~I ) _ = _ ( ( ~r_1 ~r_2 + ~x ~r_2 ) + ~I ) _ = _ ( ~r_1 ~r_2 + ~I ) , _ since _ ~x ~r_2 &in. ~I

also _ ( ~r_2 + ~I ) ( ~r_1' + ~I ) _ = _ ( ~r_2 ~r_1' + ~I ) _ = _ ( ~r_2 ( ~r_1 + ~x ) + ~I ) _ = _ ( ( ~r_2 ~r_1 + ~r_2 ~x ) + ~I ) _ = _ ( ~r_2 ~r_1 + ~I ) , _ since _ ~r_2 ~x &in. ~I

Multiplication is associative: _ #( ( ~r_1 + ~I ) ( ~r_2 + ~I ) #) ( ~r_3 + ~I ) _ = _ ( ~r_1 + ~I ) #( ( ~r_2 + ~I ) ( ~r_3 + ~I ) #)

So ~R ./ ~I is a (commutative) ring with these operations, known as the #~{quotient ring} or ~R by ~I.

If ~R and ~S are rings, then a function _ ~f #: ~R -> ~S _ is a #~{(ring) homomorphism} _ if

~f ( ~x + ~y ) _ = _ ~f ( ~x ) + ~f ( ~y ) , _ and _ ~f ( ~x ~y ) _ = _ ~f ( ~x ) ~f ( ~y ) , _ &forall. ~x, ~y &in. ~R.

If _ ~f #: ~R -> ~S _ is a homomorphism, then _ Im ~f _ is a subring of ~S, and ker ~f _ = _ \{ ~x &in. ~R | ~f ~x = 0 \} _ is a subring of ~R.

[ ker ~f _ is the (group) kernel of ~f considered as a homomorphism of the additive groups ( ~R , + ) and ( ~S , + ) .]

A #~{(ring) isomorphism} is a bijective homomorphism.

If ~R is a ring, then any non-zero element ~x, for which there exists non-zero element ~y, such that _ ~x ~y _ = _ 0 , _ or _ ~y ~x _ = _ 0 , _ is called a #~{divisor of zero}. _ [ In which case ~y is also a divisor of zero ]

A commutative ring with no divisors of zero is called an #~{integral domain}.

A ring ~R is a #~{division ring} if _ ( ~R #\ \{ 0 \} , # ) _ is a group.

Example: _ \{ 0, 1, 2, 3, 4 | (mod 5) \} is a division ring - see multiplication table in ring examples above.

A #~{field} is a commutative division ring. [ i.e. _ ( ~R #\ \{ 0 \} , # ) _ is an abelian group.]

Note that a division ring can have no divisors of zero ( otherwise ~R #\ \{ 0 \} would not be closed under multiplication ), but it is not necessarily an integral domain, as it would also have to be commutative, in which case it would be a field. So a field is a type of integral domain.

__Proposition:__ Any integral domain can be "embeded" in a field, in the sense that we can construct a field and an injective homomorphism from the integral domain into the field.

Any integral domain can be embedded in a field, in the sense that we can construct a field, and an injective homomorphism from the domain to the field

If ~R is a ring, let ~R_0 = ~R #\ \{ 0 \}. Consider the relation on ~R # ~R_0 : _ ( ~x_1 , ~y_1 ) ~ ( ~x_2 , ~y_2 ) _ if _ ~x_1 ~y_2 = ~x_2 ~y_1 . _ This is an equivalence relation, it is reflexive and symmetric (obvious), and transative: if _ ~x_1 ~y_2 = ~x_2 ~y_1 _ and _ ~x_2 ~y_3 = ~x_3 ~y_2 , _ then _ ~x_1 ~y_2 ~x_2 ~y_3 = ~x_2 ~y_1 ~x_3 ~y_2 _ => _ ~x_1 ~y_3 ( ~y_2 ~x_2 ) = ~y_1 ~x_3 ( ~y_2 ~x_2 ) _ => _ ~x_1 ~y_3 = ~y_1 ~x_3 _ [ as we can "cancel" in an integral domain.]

Write _ [ ~x , ~y ] _ for the equivalence class of ( ~x , ~y ). Define

- addition: _ [ ~x_1 , ~y_1 ] + [ ~x_2 , ~y_2 ] _ = _ [ ~x_1 ~y_2 + ~x_2 ~y_1 , ~y_1 ~y_2 ]
- multiplication: _ [ ~x_1 , ~y_1 ] # [ ~x_2 , ~y_2 ] _ = _ [ ~x_1 ~x_2 , ~y_1 ~y_2 ]

Let _ ( ~x_1 , ~y_1 ) ~ ( ~x_1' , ~y_1' ) _ and _ ( ~x_2 , ~y_2 ) ~ ( ~x_2' , ~y_2' ) _ then

- addition is well-defined: _ [ ~x_1' , ~y_1' ] + [ ~x_2' , ~y_2' ] _ = _ [ ~x_1' ~y_2' + ~x_2' ~y_1' , ~y_1' ~y_2' ]

Now ( ~x_1' ~y_2' + ~x_2' ~y_1' ) ~y_1 ~y_2 _ = _ ~x_1' ~y_1 ~y_2 ~y_2' + ~x_2' ~y_2 ~y_1 ~y_1' _ = _ ~x_1 ~y_1' ~y_2 ~y_2' + ~x_2 ~y_2' ~y_1 ~y_1' _ = _ ( ~x_1 ~y_2' + ~x_2 ~y_1 ) ~y_1' ~y_2'

so _ ( ~x_1' ~y_2' + ~x_2' ~y_1' , ~y_1' ~y_2' ) _ ~ _ ( ~x_1 ~y_2 + ~x_2 ~y_1 , ~y_1 ~y_2 )

_ - multiplication is well-defined: _ [ ~x_1' , ~y_1' ] # [ ~x_2' , ~y_2' ] _ = _ [ ~x_1' ~x_2' , ~y_1' ~y_2' ]

~x_1' ~x_2' ~y_1 ~y_2 _ = _ ~x_1' ~y_1 ~x_2' ~y_2 _ = _ ~x_1 ~y_1' ~x_2 ~y_2' _ = _ ~x_1 ~x_2 ~y_1' ~y_2' _ so _ ( ~x_1' ~x_2' , ~y_1' ~y_2' ) _ ~ _ ( ~x_1 ~x_2 , ~y_1 ~y_2 )

Note: _ any ~a &in. ~R_0 , _ ( ~a ~x , ~a y ) _ ~ _ ( ~x , ~y )

With these operations the set of equivalence classes is a field

- addition is associative: _ ( [ ~x_1 , ~y_1 ] + [ ~x_2 , ~y_2 ] ) + [ ~x_3 , ~y_3 ] _ = _ [ ~x_1 ~y_2 + ~x_2 ~y_1 , ~y_1 ~y_2 ] + [ ~x_3 , ~y_3 ]

= _ [ ~x_1 ~y_2 ~y_3 + ~x_2 ~y_1 ~y_3 + ~x_3 ~y_1 ~y_2 , ~y_1 ~y_2 ~y_3 ] _ etc. - [ 0 , ~y ] is the zero element, _ ( 0 , ~y ) ~ ( 0 , ~y' ) , _ any _ ~y , ~y' &in. ~R_0 , _ [ 0 , ~y ] + [ ~x_1 , ~y_1 ] _ = _ [ 0 + ~x_1 ~y , ~y_1 ~y ] _ = _ [ ~x_1 , ~y_1 ] .
- additive inverse: _ [ ~x_1 , ~y_1 ] + [ -~x_2 , ~y_1 ] _ = _ [ ~x_1 ~y_1 - ~x_1 ~y_1 , ~y_1 ~y_1 ] _ = _ [ 0 , ~y_1 ~y_1 ]
- addition is commutative.
- multiplication is associative: _ ( [ ~x_1 , ~y_1 ] # [ ~x_2 , ~y_2 ] ) # [ ~x_3 , ~y_3 ] _ = _ [ ~x_1 ~x_2 ~x_3, ~y_1 ~y_2 ~y_3 ]
- any ~x &in. ~R_0 , _ [ ~x , ~x ] is the unit element, _ ( ~x , ~x ) ~ ( ~x' , ~x' ) , _ any _ ~x , ~x' &in. ~R_0 , _ [ ~x , ~x ] # [ ~x_1 , ~y_1 ] _ = _ [ ~x ~x_1 , ~x ~y_1 ] _ = _ [ ~x_1 , ~y_1 ].
- inverse: _ for _ ~x != 0 _ ( i.e. [ ~x , ~y ] is not the zero element ) _ then _ [ ~x , ~y ] # [ ~y , ~x ] _ = _ [ ~x ~y , ~x ~y ]
- multiplication is commutative.
- distributive: _ ( [ ~x_1 , ~y_1 ] + [ ~x_2 , ~y_2 ] ) # [ ~x_3 , ~y_3 ] _ = _ [ ~x_1 ~y_2 + ~x_2 ~y_1 , ~y_1 ~y_2 ] # [ ~x_3 , ~y_3 ]

= _ [ ~x_1 ~x_3 ~y_2 + ~x_2 ~x_3 ~y_1 , ~y_1 ~y_2 ~y_3 ] _ = _ [ ~x_1 ~x_3 ~y_2 ~y_3 + ~x_2 ~x_3 ~y_1 ~y_3 , ~y_1 ~y_2 ~y_3 ~y_3 ] , _ (multiplying by unit element _ [ ~y_3 , ~y_3 ] )

= _ [ ~x_1 ~x_3 , ~y_1 ~y_3 ] + [ ~x_2 ~x_3 , ~y_2 ~y_3 ] _ = _ [ ~x_1 , ~y_1 ] # [ ~x_3 , ~y_3 ] + [ ~x_2 , ~y_2 ] # [ ~x_3 , ~y_3 ] , _ etc.

Consider the mapping: _ ~f #: ~R -> ~R # ~R_0 ./ ~ , _ given by _ ~f ( ~x ) _ = _ [ ~x , 1 ] . _ ~f _ is a homomorphism:

- ~f ( ~x + ~y ) _ = _ [ ~x + ~y , 1 ] _ = _ [ ~x , 1 ] + [ ~y , 1 ]
- ~f ( ~x ~y ) _ = _ [ ~x ~y , 1 ] _ = _ [ ~x , 1 ] # [ ~y , 1 ]

~f _ is injective: _ ~f ( ~x ) = ~f ( ~y ) _ => _ [ ~x , 1 ] = [ ~y , 1 ] _ => _ ( ~x , 1 ) ~ ( ~y , 1 ) _ => _ ~x # 1 = ~y # 1 _ => _ ~x = ~y .

For any element ~x in a commutative ring ~R, the set _ ~R~x #:= \{ ~r ~x | ~r &in. ~R \} _ is an ideal of ~R, called the #~{principal ideal} generated by ~x.

A #~{principal ideal ring} is a (commutative) ring in which all ideals are principal ideals.