Let $\mathbb{F}$ - field,
A = $\begin{pmatrix}
\mathbb{F} & \mathbb{F}\\
0& 0
\end{pmatrix}$,
B = $\begin{pmatrix}
\mathbb{F} & 0\\
\mathbb{F} & 0
\end{pmatrix}$.
I know that A and B are not isomorphic $\mathbb{F}$-algebras.
But will they isomorphic if I join 1 to $\mathbb{F}$-algebras:
$ A \oplus \mathbb{F} \bullet 1 \cong B \oplus \mathbb{F} \bullet 1$?