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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

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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
    If you do not know what the radical is, you can simply show by hand that if $a\in A$ iis such that at least one of $xa$ or $ya$ is not-zero, then the ideal generated by $a$ is not simple.2012-07-11
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    using the basis in the answer, you can identify $A$ with $End(A)$, and you can now check the natural bilinear form given by $tr(ab)$ where $a,b\in A=End(A)$ is non-degenerate and symmetric.2012-07-14
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    Hmm, that part of the question was not there when I wrote the answer.2012-07-14
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    @Mariano: Sorry, that was my fault. Inadvertently I added the second part to my question before having noticed your answer.2012-07-21