Let $\pi:E\rightarrow M$ be a smooth vector bundle and $p\in M$. Consider $$I_p(M)=\{f\in C^\infty(M): f(p)=0\},$$ and $\Gamma(E)$ the $C^\infty(M)$-module of smooth sections over $E$. Notice $I_p(M)$ is an ideal of $C^\infty(M)$. Can anyone tell me what the set $I_p^{k+1}(p)\cdot \Gamma(E)$ is?
This question arose when I was trying to learn about jet bundles, the reference I was following
http://vmm.math.uci.edu/PalaisPapers/SASIT.pdf (pg. 56, section Jet bundles)
didn't say what that set was and once I don't have much knowledge in algebra I got lost..
Looking at the source, it appears that there is a typographical error and what is meant is in fact
$$ I_p^{k+1}(M) \cdot \Gamma(E) $$
We have that $\Gamma(E)$ is the $C^\infty(M)$-module of smooth sections, and $I_p^{k+1}(M)\subseteq C^\infty(M)$. So a reasonable interpretation is that
$$ I_p^{k+1}(M) \cdot \Gamma(E) := \left\{ \sigma \in \Gamma(E) \middle| \sigma = f \gamma, f\in I_p^{k+1}(M), \gamma\in \Gamma(E)\right\} $$
Note that in the source (Palais' Seminar on Atiyah-Singer...) the object above is given a name $Z_p^k$ and the $k$ jet at $p$ is defined to be $\Gamma(E) / Z_p^k$, which is just saying that two smooth sections in represent the same $k$-jet if they differ by something in $Z_p^k$, which is something that vanishes to order $k+1$.