$ \mathbb{Z}_p\mathbb{Z}_p[v] $-Additive cyclic codes of length $ (\alpha,\beta) $ can be viewed as $ R[x] $-submodules of $ \mathbb{Z}_p[x]/(x^\alpha-1)\times R[x]/(x^\beta-1) $, where $ R = \mathbb{Z}_p+v\mathbb{Z}_p $ with $ v^2 = v $. In this paper, we determine the generator polynomials and the minimal generating sets of this family of codes as $ R[x] $-submodules of $ \mathbb{Z}_p[x]/(x^\alpha-1)\times R[x]/(x^\beta-1) $. We also determine the generator polynomials of the dual codes of $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes. Some optimal $ \mathbb{Z}_p\mathbb{Z}_p[v] $-linear codes and MDSS codes are obtained from $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes. Moreover, we also get some quantum codes from $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes.

Let $ R = \mathbb{F}_q[u,v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle $ be a finite non-chain ring, where $ q $ is an odd prime power and $ u^{2} = 1 $, $ v^2 = v $, $ uv = vu $. In this paper, we construct new non-binary quantum codes from ($ \alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R $. We give the structure of ($ \alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R $ and obtain self-orthogonal codes over $ \mathbb{F}_q $ by Gray map. By using Calderbank-Shor-Steane (CSS) construction and Hermitian construction from dual-containing ($ \alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R $, some new non-binary quantum codes are obtained.