Weighted EGZ Constant for p-groups of rank 2

Let $G$ be a finite abelian group of exponent $n$, written additively, and let $A$ be a subset of $\mathbb{Z}$. The constant $s_A(G)$ is defined as the smallest integer $\mathrm{\ell }$ such that any sequence over $G$ of length at least $\mathrm{\ell }$ has an $A$-weighted zero-sum of length $n$ and ${\eta }_A(G)$ defined as the smallest integer $\mathrm{\ell }$ such that any sequence over $G$ of length at least $\mathrm{\ell }$ has an $A$-weighted zero-sum of length at most $n$. Here we prove that, for $\alpha \ge \beta$, and $A=\left{x\in$\mathbb{N}$; : ; 1 \le a \le p^{\alpha} ; \mbox{ and }; \gcd(a, p) = 1\right }$,wehave$s_{A}($\mathbb{Z}${p^{\alpha}}\oplus $\mathbb{Z}${p^\beta}) = \eta_A($\mathbb{Z}${p^{\alpha}}\oplus $\mathbb{Z}${p^\beta}) + p^{\alpha}-1 = p^{\alpha} + \alpha +\beta$andclassifyalltheextremal$A$-weighted zero-sum free sequences.