## A note on the countable extensions of separable $p^{\omega +n}$-projective abelian $p$-groups

*P. Danchev*

**Address****.**

13, General Kutuzov Street, block 7, floor 2, flat 4, 4003 Plovdiv, Bulgaria

**E-mail.** pvdanchev@yahoo.com**
**

**Abstract****.**

It is proved that if $G$ is
a pure $p^{\omega + n}$-projective
subgroup of the separable abelian

$p$-group $A$ for
$n\in {\Bbb N}\cup \{0\}$
such that $|A/G|\leq \aleph_0$, then $A$ is $p^{\omega+n}$-projective as well. This generalizes
results due to Irwin-Snabb-Cutler
(Comment\. Math\. Univ\. St\. Pauli,

1986) and the
author (Arch.\ Math.\
(Brno), 2005).

**AMSclassification**.
20K10.

Keywords. Countable
extensions, separable groups, $p^{\omega+n}$-projective
groups.