# Notes on countable extensions of $p^{\omega +n}$-projectives

## Peter Danchev

Address: 13, General Kutuzov Street, bl. 7, floor 2, flat 4 4003 Plovdiv, Bulgaria

E-mail: pvdanchev@yahoo.com

Abstract: We prove that if $G$ is an Abelian $p$-group of length not exceeding $\omega $ and $H$ is its $p^{\omega +n}$-projective subgroup for $n\in {\mathbb{N}} \cup \lbrace 0\rbrace $ such that $G/H$ is countable, then $G$ is also $p^{\omega +n}$-projective. This enlarges results of ours in (Arch. Math. (Brno), 2005, 2006 and 2007) as well as a classical result due to Wallace (J. Algebra, 1971).

AMSclassification: Primary: 20K10; Secondary: 20K25.

Keywords: abelian groups, countable factor-groups, $p^{\omega +n}$-projective groups