A Boolean Ring on a Power Set

Detailed Solution

The multiplication in this ring is defined by intersection: $$ A\otimes B=A\cap B. $$ For every subset $A\subseteq X$, the intersection of $A$ with itself is $A$. Therefore, $$ A\otimes A=A\cap A=A. $$ Hence every element of the ring is idempotent under multiplication. This proves that option B is correct. We verify the other options. Option A is false. The multiplicative identity is $X$, because $$ A\cap X=A $$ for every $A\subseteq X$. A nonempty subset $A$ is a unit only if there exists $B\subseteq X$ such that $$ A\cap B=X. $$ This requires $A=X$. Thus $X$ is the only unit, so not every nonempty subset is a unit. Option C is false. Intersection is commutative: $$ A\otimes B=A\cap B=B\cap A=B\otimes A. $$ Option D is false. The characteristic of a ring is the least positive integer $n$ for which $n\cdot 1_R=0_R$. Here, the additive identity is $\varnothing$, and the multiplicative identity is $X$. Since addition is symmetric difference, $$ X\oplus X=\varnothing. $$ Thus $2\cdot 1_R=0_R$, while $X\neq\varnothing$. Hence the characteristic is $2$, not $3$. Therefore, the correct option is $\boxed{\mathrm{B}}$.