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So, I can write down the definition of the socle of a module but I'm having trouble actually putting it in my brain. To understand things I'm trying to write down the radical and socle series of the lower triangular matrices. In general, if $A$ is a finite dimensional $k$-algebra is it the case that $soc^2 A$ is the stuff annihilated by $rad^2 A$ or is there more?

Let $T_n(k)$ be the lower triangular matrices over a field $k$. It is easily seen that

$$rad T_n(k) = \begin{pmatrix}0 & 0 & 0\\ a & 0 & 0\\ b & c & 0 \end{pmatrix}$$

$$rad^2 T_n(k) = \begin{pmatrix}0 & 0 & 0\\ 0 & 0 & 0\\ b & 0 & 0 \end{pmatrix}$$

and $rad^3 T_n(k)=0$. Now, the socle is the stuff annihilated by the radical so

$$soc T_n(k) = \begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & 0 \\ a & b & c\end{pmatrix} $$

the stuff annihilated by $rad^2(u)$ is $$ \begin{pmatrix}a & 0 & 0\\ b & c & 0\\ 0 & x & y \end{pmatrix}$$ and

$$T_n(k)/soc T_n(k) = \begin{pmatrix}a & 0 & 0\\ b & c & 0\\ 0 & 0 & 0 \end{pmatrix}$$

and the socle of this is the sum of the simple submodules of that quotient (which I'm having trouble determining) but it seems these things don't line up? What's going on?

RhythmInk
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    There's a mistake: $\mathrm{rad}^2,T_3(k)$ annihilates matrices of the form $\begin{pmatrix}0&0&0\x&y&0\a&b&c\end{pmatrix}$. This is therefore $\mathrm{soc}^2,T_3(k)$, it contains $\mathrm{soc},T_3(k)$, the quotient is $$\mathrm{soc}^2,T_3(k)/\mathrm{soc},T_3(k) = \begin{pmatrix}0&0&0\x&y&0\0&0&0\end{pmatrix},$$ and this is the socle of $T_3(k)/\mathrm{soc},T_3(k)$. – Andrew Hubery Apr 03 '19 at 07:23
  • The definition is correct: $\mathrm{soc}^2,A$ is precisely those elements annihilated by $\mathrm{rad}^2,A$. – Andrew Hubery Apr 03 '19 at 07:29
  • Is that true in general or only for finite dimensional algebras? – RhythmInk Apr 03 '19 at 11:47
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    I was referring to the question, which asked about finite dimensional algebras. It holds more generally for semiprimary rings, which are those rings $A$ with nilpotent Jacobson radical $J$ such that $A/J$ is semisimple. It fails in general, for example $\mathbb Z$ has zero radical and zero socle. – Andrew Hubery Apr 03 '19 at 13:12

1 Answers1

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As mentioned in a comment, check your computations again. The radical and socle series of the lower triangular matrices (considered as a right module over itself) should look like this:

radical series: $$ \begin{bmatrix}\cdot&0&0\\\cdot&\cdot&0\\\cdot&\cdot&\cdot\\\end{bmatrix}\supseteq \begin{bmatrix}0&0&0\\\cdot&0&0\\\cdot&\cdot&0\\\end{bmatrix}\supseteq\begin{bmatrix}0&0&0\\0&0&0\\\cdot&0&0\\\end{bmatrix}\supseteq\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\\\end{bmatrix} $$

socle series: $$ \begin{bmatrix}0&0&0\\0&0&0\\0&0&0\\\end{bmatrix}\subseteq \begin{bmatrix}0&0&0\\0&0&0\\\cdot&\cdot&\cdot\\\end{bmatrix}\subseteq \begin{bmatrix}0&0&0\\\cdot&\cdot&0\\\cdot&\cdot&\cdot\\\end{bmatrix}\subseteq \begin{bmatrix}\cdot&0&0\\\cdot&\cdot&0\\\cdot&\cdot&\cdot\\\end{bmatrix}$$

(The obvious generalization to $n\times n$ lower triangular matrices is hopefully self-evident, although inconvenient to write out.)

rschwieb
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