1. ## Help with proof

If X is an infinite set, then the family of all finite subsets of X forms a subring of the Boolean Ring $\displaystyle \beta (X)$.

The book says this statement is true, and we have to prove it... but i think it is false

i think the unity is X, and it is NOT finite, and it is an ideal????

2. Originally Posted by ux0
If X is an infinite set, then the family of all finite subsets of X forms a subring of the Boolean Ring $\displaystyle \beta (X)$.

The book says this statement is true, and we have to prove it... but i think it is false

i think the unity is X, and it is NOT finite, and it is an ideal????
you haven't got any response yet because you didn't define $\displaystyle \beta(X)$ for us.

3. The Boolean ring $\displaystyle \beta (X)$ has elements which are subsets of $\displaystyle X$, with multplication defined as their intersection: $\displaystyle AB = A \cap B$, and with addition defined as their symmetric difference: $\displaystyle A+B= (A-B) \cup (B-A)$.. Also the identity is $\displaystyle X$

4. well, it depends on how you define a subring: generally a subring doesn't have to have a multiplicative identity but if it has, then it should be equal to the identity element of the ring.

in your example, your subring doesn't have the identity element and that is fine.

5. we define a subring as a subset R of a commutative ring S is a subring of S if

$\displaystyle 1 \in R$

if $\displaystyle a,b \in R$, then $\displaystyle a-b \in R$

if $\displaystyle a,b \in R$, then $\displaystyle ab \in R$

The book said it is true though that is the problem... i shouldn't have a counter example.