Boolean ring

Quote:

Originally Posted by petter

Prove that in a Boolean ring with http://alt2.mathlinks.ro/latexrender...eee25ea5d3.gif elements there exists http://alt1.mathlinks.ro/latexrender...8721dd1e29.gif s.t. http://alt1.mathlinks.ro/latexrender...d572d3b34d.gif for all distinct http://alt1.mathlinks.ro/latexrender...9b9206a86b.gif.

since the ring $R$ is Boolean, it is unitary, commutative and has no non-zero nilpotent element. hence $R$ is the direct sum of $n$ copies of $\mathbb{F}_2.$ now for any $1 \leq i \leq n$ let $a_i=(x_{i1}, \cdots , x_{in}) \in R,$

where $x_{ii}=1$ and $x_{ik}=0, \ \forall k \neq i.$ obviously $a_ia_j=0, \ \forall i \neq j. \ \ \Box$