Proving that a subset of a ring $R$ is a subring In this example, $R$ is a ring with unity $1$, with $a\in R$ having the property $a^2=a$ (making it a Boolean ring). I know every Boolean ring is of characteristic 2 since: $a+a=(a+a)^2=a^2+a^2+a^2+a^2=a+a+a+a \implies a+a=0$ The subset is defined as $aRa\subseteq R$ by $aRa=${$ara | r\in R$}. How would I go about proving, or disproving, that $aRa$ is a subring of $R$ given the subset? Would $aRa$ contain the same unity element $1$ Excuse the lengthy question, rings are proving to be a particularly pertinent frustration for me in Abstract Algebra.

Boolean rings are commutative (see below for a proof) so that $ara=a^2r=ar$. So $aRa$ will only contain the identity if $a$ is a unit.

Also $a(1-a)=a-a^2=a-a=0$ showing that $a\
eq1$ implies that $a$ is not a unit.

Final conclusion $aRa$ is a subring of $R$ if and only if $a=1$.

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proof of commutativity:

$a+b=(a+b)^2=a^2+ab+ba+b^2=a+ab+ba+b$ so that $ab+ba=0=ab+ab$. So $ba=ab$.