0
$\begingroup$

A module $M$ is called H-supplemented if for every submodule $N$ of $M$ there exists a direct summand $D$ of $M$ such that $M = N + X$ if and only if $M = D + X$ for every submodule $X$ of $M$. A module $M$ has $D2$ if $A\leq M$ such that $M/A$ is isomorphic to a direct summand of $M$, then $A$ is a direct summand of $M$. The question is: Does H-supplemented imply D2?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
3
$\begingroup$

Isn't the $\mathbb{Z}$-module $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}$ $H$-supplemented but not $D2$?

Improve this answer
$\endgroup$
2
  • $\begingroup$ Thanks for you reply, I don't see why it's D2. Could you please explain? $\endgroup$
    – Hajer Bagdady
    Commented May 31, 2020 at 0:35
  • 1
    $\begingroup$ @HajerBagdady The claim is that it's not D2. Take $A = (\Bbb Z/2\Bbb Z, 2\Bbb Z/4\Bbb Z)$. If this was a direct summand the module would be isomorphic to $(\Bbb Z/2)^3$. $\endgroup$
    – mme
    Commented May 31, 2020 at 0:49

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .