Let
![$\mathcal{R}={\Bbb Z}_p+u{\Bbb Z}_p+u^2{\Bbb Z}_p$](img26.png)
be a
commutative ring with
![$u^3=u$](img27.png)
and
![$p$](img28.png)
is an odd prime. The
![${\Bbb
Z}_p\mathcal{R}$](img29.png)
-additive cyclic codes can be considered as
![$\mathcal{R}[x]$](img30.png)
-submodules of
![$\frac{{\Bbb Z}_p[x]}{<x^\alpha -1>}
\times\frac{\mathcal{R}[x]}{<x^\beta -1>}$](img31.png)
, for some positive
integers
![$\alpha$](img32.png)
and
![$\beta$](img33.png)
. In this paper, we study the algebraic
structure of
![${\Bbb
Z}_p\mathcal{R}$](img29.png)
-additive cyclic codes of length
![$(\alpha, \beta)$](img34.png)
. To do this, we determine their generator
polynomials and minimal generating sets. Moreover, we discuss the
duality of the
![${\Bbb
Z}_p\mathcal{R}$](img29.png)
-additive cyclic codes and
obtain their generator polynomials. We also study the structure of
additive constacyclic codes and quantum codes over
![${\Bbb
Z}_p\mathcal{R}$](img29.png)
.