Show that $\mathbb{Z}/(p^n),p$ a prime $n\ge 0$, regarded as $\mathbb{Z}$-module is not a direct sum of any two non-zero submodules. Does this hold for $\mathbb{Z}/(n)$ for other positive nteger ???
Please help anyone .
Asked
Active
Viewed 76 times
3
Aritra Roy
- 109
-
$\mathbb Z$-modules are just Abelian groups. Do you possibly already know this result from group theory? – Aug 26 '20 at 15:06
-
yes i know that – Aritra Roy Aug 26 '20 at 15:17
1 Answers
2
A necessary condition for a module $M$ to be $X\oplus Y$, for nonzero submodules $X$ and $Y$, is that there exist nonzero submodules $X,Y$ such that $X\cap Y=\{0\}$.
Is it possible for $\mathbb{Z}/p^n\mathbb{Z}$? What are the submodules (subgroups)?
Is it possible for $\mathbb{Z}/n\mathbb{Z}$? Yes. Try with $n=6$.
egreg
- 238,574
-
for $\mathbb{Z}/n p\mathbb{Z}=\mathbb{Z}/n2\mathbb[Z} \oplus \mathbb{Z}/n 3 \mathbb{Z}$ by CRT, Subgroup of Z/p^nZ are those whose order is divisors of p^n – Aritra Roy Aug 26 '20 at 16:03