I am asked to show that the following sets are rings. And determine if they commutative.

A set that is a ring...... if you didn't know is (I am sure you did though)...

So I am asked to "show" the following sets of rings. Furthermore my prof said ...1. Closure under addition. For alla,binR, the result of the operationa+bis also inR.c[›] 2. Associativity of addition. For alla,b,cinR, the equation (a+b) +c=a+ (b+c) holds. 3. Existence of additive identity. There exists an element0inR, such that for all elementsainR, the equation0+a=a+0=aholds. 4. Existence of additive inverse. For eachainR, there exists an elementbinRsuch thata+b=b+a= 0 5. Commutativity of addition. For alla,binR, the equationa+b=b+aholds.

You have to show the rules hold, for *all* numbers inSo I have to show that the rule holds for all values in the set for...

the set.

1. Z[x], the set of all polynomials in x with integer coeeficients.

So i know already that all these rules hold for all integer values. And multiplying and dividing

any polynomial with integer coefficients together will render another polynomial with integer coefficients. So how do I show this obvious fact for all value in x?

I also have to show for...

2.The set of all matrices with rational entries

And this last one

3.Q(√2) = { a + b√2 | a, b are rationals }

Does this previous one mean the set of all rational numbers a,b such that (a+b√2)

Also for number 3 I must show that it forms a field so i must show that there is a multiplicative inverse for every non zero entry. I understand what this means, but really have no clue how to show it. :S

Any help in any amount would be so appreciated!!!!

EDIT: Another related question

What do these small C's mean?6. Make a Venn diagram illustrating the case Ac∩Bc∩C = Ac∩B∩Cc = ∅, B ≠ C, and B ∩ C ≠ ∅.