Question 867410
1)The properties of a Ring include that it is a Group under addition whose identity is 0 and inverse notation is -r where r is member of the Group.
In a Group under addition every element (a) has an inverse such that
a + b = b + a = 0 and b is a's inverse denoted by -a
we are given a + a = 0, a = 0 which implies that a is its own inverse
note that 0, the additive identity element is unique and a required member of the additive Group, under closure property a + a is a member of the additive Group
2)An idempotent element a of the Ring is defined as
a^2 = a^1, thus the element is idempotent under the Ring's multiplication and moreover we can say a^1 = a^2 = a^3 = a^4 = .....a^n or any positive integer n
if every element of a ring R is an idempotent, we are dealing with a Boolean Ring
a+a = (a+a)^2 = a^2 + a^2 +a^2 +a^2 = a +a +a +a
now subtract a+a from both sides of the =
0 = a +a
addition table
row/ 0   1   x   1+x
column      
0    0   1   x   1

1    1   0  1+x  x

x    x  1+x  0   1

1+x  1+x  x   1   0
multiplication table
row/ 0   1   x   1+x
column     
0    0   0   0   0  

1    0   1   x   1+x

x    0   x   x    0

1+x  0   x   0   1+x