Let $n\in \mathbb{N}$ and $k$ be an arbitrary field.
Is the socle of the algebra $k[x,y]/\langle x^2,y^{n+2}\rangle$ isomorphic to $k$?
Is $k[x,y]/\langle x^2,y^{n+2}\rangle$ a symmetric algebra or a Frobenius algebra or a self-injective algebra?
I would be very grateful for an answer.