spectrum of a general algebra

Question

For unital Algebras $A$, the Spectrum of an element $x \in A$ is defined to be $$ Sp_A(x) := \{ \lambda \in \mathbb{C}: x - \lambda * e \text{ is not invertible} \} $$ According to my textbook, there is a way we can extend this definition to non-unital Algebras: Let $A$ be an algebra, then define $A_I := A \times \mathbb{C}$ with the operation: $(x,\lambda) · (y, \mu) = (xy + \lambda + \mu, \lambda \mu)$. Then, $A_I$ is a unital Algebra with unit $e:= (0,1)$ and we define $$ Sp_A(x) = Sp_{A_I}(x) $$ My problem is, that $x$ as an element of $A$ is not a well defined element of $A_I$; how should I understand this? Any help is apreciated