Ring, Commutative Ring, Integral Domain, Field

I did not do well on my assignment. This is what I have learned after reading the instructor's solution

To prove R is a ring we need
a) R is Closed under addition : It is neccesary to show that if a,b is in R then a+b is in R.
b) Addition is Associative : (a+b)+c = a+(b+c)
c) Addition is Commutative : a+b=b+a
d) There is an Addition Identity "0" such that 0+a=a for all a in R

e) R is Closed under multiplication : It is neccesary to show that if a,b is in R then a.b is in R
f) Multiplication is associative : a(b.c) =(a.b).c
g)Multiplication is Commutative : a.b=b.a
h) Distribution Law is satisfied : for all a,b,c : a.(b+c) = a.b+a.c
i) There is an multiplicative identity "1" such that 1.a=a for all a in R

If c) and g) are satisfied then R is a Commutative Ring. Anyway, to prove R is a commutative Ring, you still need to prove It is a ring


To prove R is an Integral Domain :
1) Prove R is a Ring
2) That 0 is the only zero divisor in R . ( a is a zero divisor if there is a value b in R such that b not equal 0 and a.b =0) . Hence , the way you should try is " if a,b is in R such that a is not equal 0 and b is not equal 0 then a.b is not equal 0 as well "

To prove R is a Field :
1) Prove R is a Integral Domain
2) Every nonzero element of R is a unit. ( let a in R, a is a unit if there is b in R such that a.b =1 in