A set,R,with two operations, + and *, is a ring if the following properties are shown to be true:

6. Associative property of multiplication: for everyq, s, andt inR,q *(s * t) = (q * s) * t

7. Commutative property of addition: for allsand t inR,s + t = t + s

8. Left distributive property of multiplication over addition:for everyq, s, andtin R,q *(s + t) = q * s + q * t

9. Right distributive property of multiplication over addition:for everyq, s, and t inR, (s + t) * q = s * q + t * q

Consider below the setM(2Z) of 2 2 matrices with even integer entries. The notation "2Z" denotes the set of even integers:

M(2Z) = {[a b c d] |a,b,c, andd 2Z}

Recall that matrix addition and multiplication are defined as follows for 2 2 matrices:

[a b c d] +[ w x y z] =[a +w b + x c +y d +z]

[ a b c d] *[ w x y z] =[ aw +by ax + bz cw +dy cx+dz]

The even integers 2Z form a ring with the usual operations of integer addition and multiplication. Given this fact, you are asked to prove thatM(2Z) also has properties of a ring in part A (properties 6-9) . Each step of each proof must be justified using the definitions of the matrix operations or an appropriate property from the ring of even integers .