GROUPS, ABELIAN GROUPS, RINGS, & FIELDS

GROUPS 

Definition: A group G is a non empty set with one binary operation multiplication (⋅) denoted as (G, ⋅)  where the following properties hold

1. Closure: For all   a, b ∈ G,  a⋅b ∈ G.
   
   
2. Associativity: For all a, b , c ∈ G, we have a⋅(b⋅ c) = (a⋅b)⋅ c
   
   
3. Identity: There exists identity element d ∈ G such that d⋅ a =  a⋅d =a  for all a ∈ d
   
   
4. Inverses: For each a ∈ G there exists an inverse element a  a-1 ∈ G such that  a⋅a-1 = a-1⋅a-1=d
   
   

Abelian Groups Definition: A group G  is an abelian group if multiplication commutes. For all a, b ∈ G, we have a⋅b = b⋅ a

RINGS (Non-commutative)

1. Definition: A ring R (noncommutative) is a  set with two binary operations addition (+) and multiplication (⋅) could be denoted as (R, +,⋅) where the following properties hold
   
   
   1. Closure (under +, ⋅ ):                                                                                                                                           Addition and Multiplication are closed i.e sum and product of all elements are in R.  For all a, b ∈ R, a + b ∈ R and a⋅b ∈ R .
      
      
   2. Associativity  (for +, ⋅):                                                                                                                                   Addition and Multiplication are associative. For all a, b, c ∈ R,   a + (b + c) = (a + b) + c and a⋅(b⋅ c) = (a⋅b)⋅ c.
      
      
   3. Additive Identity (for +):                                                                                                                                     R has an additive identity element namely 0. For all a ∈ R, a + 0 = 0 + a.
      
      
   4. Commutativity (for +): Addition is commutative. If a, b ∈ R  a + b = b + a.  
   5. Additive Inverse (for +):                                                                                                                                       R has an additive inverse element -a. For each a in R there exists an element -a ∈ R such that a +(-a)= a +(-a) = 0.
      
      
   6. Distributivity: ( ⋅ over +)                                                                                                                                     Multiplication distributes over addition:  If a, b, c ∈ R  a⋅(b + c) = a⋅b + a⋅c
      
      
   
   Examples: Matrices                                             

Commutative Ring

A Ring R is a Commutative  if multiplication is commutative.  That is for all  a,b ∈ R  a ⋅b = b ⋅ a. Note that the commutativity of commutative rings applies to multiplication since all rings are already  commutative under addition.  

Commutative Ring with Identitiy

A ring R is  a ring with Identity if there is a multiplicative identity element in R. This element is 1. A ring with identity is one such that there exists a 1 in R such that 
                                                                        a · 1 =1 · a = a   for all a∈R

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FIELDS

Rings Based Definition: A Field  F is a nonzero commutative ring where all its nonzero elements have a multiplicative inverse. It can also be thought of as a ring whose nonzero elements form an abelian(commutative) group.

Stand alone Definition: A Field F is a set with two binary operations addition (+) and multiplication (⋅) which could be denoted as (F, +,⋅) where the following properties hold

1. Closure (under + & ⋅ ):                                                                                                                                             Addition and Multiplication are closed i.e sum and product of all elements of F are in F.  For all a, b ∈ R, a + b ∈ F and a⋅b ∈ F .
   
   
2. Associativity  (for +& ⋅):                                                                                                                                   Addition and Multiplication are associative. For all a, b, c ∈ F,   a + (b + c) = (a + b) + c and a⋅(b⋅ c) = (a⋅b)⋅ c
   
   
3. Commutativity (for + & ⋅): Addition and multiplication are commutative. For all a, b ∈ F,  a + b = b + a and  a ⋅ b = b ⋅ a.
4. Additive and Multiplicative Identity elements (for + & ⋅):                                                                             F has additive and multiplicative identity elements namely 0 and 1 respectively. For all a ∈ F,  a + 0 = 0 + a = a and  a ⋅ 1 = 1 ⋅ a = a
   
   
5. Additive and Multiplicative Inverse elements (for + & ⋅):                                                                                                     F has additive and multiplicative elements denoted by  -a and a-1 respectively. For all a ∈ F,  a + (-a)= (-a) + a = 0 and  a ⋅ a-1 = a-1⋅ a = 1
6. Distributivity: ( ⋅ over +)                                                                                                                                            Multiplication distributes over addition:  If a, b, c ∈ F  a⋅(b + c) = a⋅b + a⋅c