Let $C$ be a finitely generated $R$-algebra where $R$ is unital commutative ring. Let $S$ be a simple $C$-module. Show that the annihilator of $S$ in $R$ is a maximal ideal. What I have tried is: When $C=R$ then it is a direct consequence of the simplicity of $S$ as an $R$-module. Further, since $S$ is simple then $S$ is cyclic (i.e. $S=Cy$ for nonzero $y\in S$). Consider the map $$f: C \to S$$ $$c \mapsto cy$$ Then $\ker(f)$ is the annihilator of $S$ in $C$ and $\ker(f) \cap R$ is the annihilator of $S$ in $R$. I got an idea : If we restrict the map $f: C \to S$ on $R$. Then we have an isomorphism: $f(R)=Ry \cong R/(\ker(f) \cap R)$. So to show $R/(\ker(f) \cap R)$ is maximal, it is enough to show that $Ry$ is simple as an $R$-module. Let $ry \in Ry$ be a nonzero element. Then clearly $Ry$ contains $\langle ry \rangle$ as $R$-modules. Conversely, since $ry \in Ry \subset S$ is nonzero, $\langle ry \rangle = S \supset Ry$. So $Ry = \langle ry \rangle$ for any nonzero $ry\in Ry$. Hence $Ry$ is simple and  $R/(\ker(f) \cap R)$ is maximal. Please let me know if there is ambiguity in my answer.
Let $C$ be a finitely generated $R$-algebra where $R$ is unital commutative ring. Let $S$ be a simple $C$-module. Show that the annihilator of $S$ in $R$ is a maximal ideal. What I have tried is: When $C=R$ then it is a direct consequence of the simplicity of $S$ as an $R$-module. Further, since $S$ is simple then $S$ is cyclic (i.e. $S=Cy$ for nonzero $y\in S$). Consider the map $$f: C \to S$$ $$c \mapsto cy$$ Then $\ker(f)$ is the annihilator of $S$ in $C$ and $\ker(f) \cap R$ is the annihilator of $S$ in $R$. I have shown that it is an ideal of $R$ but I have no idea to continue showing the maximality. Is it a true direction or is there anything else I should consider to show the maximality?
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Grub: reboot to specific kernel I'd like to ask for help getting grub-reboot to work.  The theory: According to  (and other sources) I should be able to choose the kernel for the next reboot, using grub-reboot. However, this does not work for me; grub always boots to the default kernel.  Here's what I tried:  $ grep GRUB_DEFAULT /etc/default/grub GRUB_DEFAULT=saved $ grep "menuentry " /boot/grub/grub.cfg  menuentry 'Ubuntu' --class ubuntu --class gnu-linux --class gnu --class os $menuentry_id_option 'gnulinux-simple-f3a807d1-3fd6-4970-b741-5382e9801060' {     menuentry 'Ubuntu, with Linux 4.4.0-43-generic' --class ubuntu --class gnu-linux --class gnu --class os $menuentry_id_option 'gnulinux-4.4.0-43-generic-advanced-f3a807d1-3fd6-4970-b741-5382e9801060' {     menuentry 'Ubuntu, with Linux 4.4.0-43-generic (recovery mode)' --class ubuntu --class gnu-linux --class gnu --class os $menuentry_id_option 'gnulinux-4.4.0-43-generic-recovery-f3a807d1-3fd6-4970-b741-5382e9801060' {     menuentry 'Ubuntu, with Linux 4.4.0-36-generic' --class ubuntu --class gnu-linux --class gnu --class os $menuentry_id_option 'gnulinux-4.4.0-36-generic-advanced-f3a807d1-3fd6-4970-b741-5382e9801060' {     menuentry 'Ubuntu, with Linux 4.4.0-36-generic (recovery mode)' --class ubuntu --class gnu-linux --class gnu --class os $menuentry_id_option 'gnulinux-4.4.0-36-generic-recovery-f3a807d1-3fd6-4970-b741-5382e9801060' {  # Version 1, using the name $ sudo grub-reboot "Ubuntu, with Linux 4.4.0-36-generic"  # Version 2, using the id $ sudo grub-reboot gnulinux-4.4.0-36-generic-advanced-f3a807d1-3fd6-4970-b741-5382e9801060  # Version 3, counting $ sudo grub-reboot 3   Neither of these commands seem to have an effect; after reboot, I always end up with the 4.4.0-43 kernel.  Note: This is on Ubuntu 16.04 Xenial. It is a server machine and I don't have access to the console during boot, so I can't look at the grub menu :( Cannot add a NOT NULL column with default value NULL in Sqlite3