Existence and uniqueness of unital ring containing a non-unital ring as maximal ideal

Question

I think the title is pretty self-descriptive: let $I$ be a non-unital ring. Is there a unique (up to isomorphism) unital ring $R(I)$, s.t. $I \subseteq R(I)$ is a maximal ideal?

Answer 1

Let $F$ be any subfield of $\Bbb C$, and $I=(x)\lhd \Bbb C[x]$.

Then the ring $R=F+I$ is a subring of $\Bbb C[x]$ with identity, $I$ is clearly an ideal of this ring, and $R/I\cong F$, so $I$ is a maximal ideal. (This matches the 'idealization' of $I$ by $F$, a.k.a. the Dorroh extension.)

For $F=\Bbb Q$ and $F=\Bbb R$, you get nonisomorphic rings, for the group of units in $\Bbb Q+I$ is countable while the group of units in $\Bbb R+ I$ is uncountable.