0
$\begingroup$

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.

1 Answers 1

1

Your algebra $A$ has the set $\mathcal B=\{x^iy^j:0\leq i<2, 0\leq j as a basis. An element $a$ of $A$ is in the socle iff $xa=ya=0$, because $x$ and $y$ generate the radical of $A$ (This last statement has to be checked of course: the ideal $I$ generated by $x$ and $y$ is nilpotent, because $x$ and $y$ are, and the quotient $A/I$ is a field, so $I$ is the radical by a well-known characterization of the radical)

You can easily find which linear combinations of the elements of $\mathcal B$ satisfy these two conditions.

  • 0
    @Mariano: Sorry, that was my fault. Inadvertently I added the second part to my question before having noticed your answer.2012-07-21