Let $\mathbb{Z}_{p}$ be the ring of residue classes modulo a prime $p$. The $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic codes of length $(\alpha,\beta)$ is identify as $\mathbb{Z}_{p}[u,v][x]$-submodule of $\mathbb{Z}_{p}[x]/\langle x^{\alpha}-1\rangle \times \mathbb{Z}_{p}[u,v][x]/\langle x^{\beta}-1\rangle$ where $\mathbb{Z}_{p}[u,v]=\mathbb{Z}_{p}+u\mathbb{Z}_{p}+v\mathbb{Z}_{p}$ with $u^{2}=v^{2}=uv=vu=0$. In this article, we obtain the complete sets of generator polynomials, minimal generating sets for cyclic codes with length $\beta$ over $\mathbb{Z}_{p}[u,v]$ and $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic codes with length $(\alpha,\beta)$ respectively. We show that the Gray image of $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic code with length $(\alpha,\beta)$ is either a QC code of length $4\alpha$ with index $4$ or a generalized QC code of length $(\alpha,3\beta)$ over $\mathbb{Z}_{p}$. Moreover, some structural properties like generating polynomials, minimal generating sets of $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive constacyclic code with length $(\alpha,p-1)$ are determined.

翻译：设$\mathbb{Z}_{p}$为模素数$p$的剩余类环。长度为$(\alpha,\beta)$的$\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-加性循环码被定义为$\mathbb{Z}_{p}[x]/\langle x^{\alpha}-1\rangle \times \mathbb{Z}_{p}[u,v][x]/\langle x^{\beta}-1\rangle$的$\mathbb{Z}_{p}[u,v][x]$-子模，其中$\mathbb{Z}_{p}[u,v]=\mathbb{Z}_{p}+u\mathbb{Z}_{p}+v\mathbb{Z}_{p}$满足$u^{2}=v^{2}=uv=vu=0$。本文分别获得了$\mathbb{Z}_{p}[u,v]$上长度为$\beta$的循环码以及长度为$(\alpha,\beta)$的$\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-加性循环码的生成多项式完整集合与最小生成集。我们证明了长度为$(\alpha,\beta)$的$\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-加性循环码的Gray像是$\mathbb{Z}_{p}$上长度为$4\alpha$且指标为$4$的QC码，或者是长度为$(\alpha,3\beta)$的广义QC码。此外，还确定了长度为$(\alpha,p-1)$的$\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-加性常循环码的生成多项式与最小生成集等结构性质。