The correspondence between maximal ideals in an algebra and it's unitalization

Question

Let $A_+$ denote the unitalization of a $\mathbb{C}$-algebra $A$ ( which is $A \oplus \mathbb{C}$ endowed with well-know multiplication rule. I know that the map $\Omega(A_+) \to \Omega(A)$, $J \mapsto J \cap A$ where $\Omega$ means the maximal ideals and $A$ is considered as it's image in $A_+$, is a bijection (almost may be we should add $A$ to the image). I am asking for some hint. I know it is most likely pretty obvious but I can't see it for some reason.

Answer 1

The map you speak of seems not always to be a bijection. You can take a $\mathbb C$ algebra with no maximal ideals, and embed it into its unitization (which must have maximal ideals.)

Or you can take something like $\mathbb C\times\mathbb C$ with trivial multiplication to be a $\mathbb C$ algebra, and that has infinitely many maximal ideals (any $1$ dimensional subspace works.) but when you take the unitization, the original rng is the unique maximal ideal of the unitization.