Question 867410: 1)Prove that if a+a=a a=0 a is an element of ring R
2) if every element of a ring R is an idempotent , prove that a+a=0 3) what is addition and multiplication table for {0,1,x,1+x} note that each element is an idempotent .
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! 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
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