We know that the socle of a module is the sum of all the simple submodules. I was checking this for the following example, but it doesn't seem to line up with the definitions since there are obviously simple modules of the form $(0,1,0)^T$ whereas the socle of the example is only the sum of $(0,0,1)^T$ Thank you! Example: Confusion over Socle
Asked
Active
Viewed 52 times
1 Answers
0
I think you are referring to the subset $\begin{bmatrix}0&0&0\\ a&b&0\\0&0&0\end{bmatrix}$ based on your attempted description of $\begin{bmatrix}0&0&0\\ 0&0&0\\ a&b&c\end{bmatrix}$ as “a sum of column vectors.”
One way to refute that is, as the link says, is that the Jacobson radical does not annihilate elements of that set. They are not as “obviously” simple as you thought.
It's true that the left socle of the lower triangular matrix ring over a field is a direct sum of copies of the set of matrices $[0,0,a]^T$, which is a simple left module. But if you think the set of matrices $[0,a,0]^T$ is a simple left $R$ module, you'll have to check again: it isn't even closed under multiplication by $R$ on the left.
rschwieb
- 153,510