I am trying to make sense of Definition 1.9 of the following paper. A graded ring is defined via a sequence of abelian groups; I outline the details below.
Let $p$ be an odd prime. Define $J_0:=Z_{(p)}$, the localisation of $\mathbb{Z}$ at the prime ideal $p\mathbb{Z}$. Also define $J_{-2}:=\mathbb{Q}_p/\mathbb{Z}_p$, the quotient of the field of $p$-adic numbers by the $p$-adic integers (Presumably only the abelian group structure is of interest at this stage..)
For each non-zero integer $k$, we let $J_{2(p-1)k-1}$ be the cyclic group of order $p^{v_p(k)+1}$, and choose a generator $\alpha_k$. So $$J_{2(p-1)k-1}=\lbrace 1,\alpha_k,\alpha_k^2,\ldots,\alpha_k^{p^{v_p(k)}}\rbrace$$ for all non-zero integers $k$.
Then the product structure is discussed. But how are the remaining abelian groups defined? For example if $p=3$, we have defined $J_{4k-1}$ for all non-zero $k$, and also $J_0$ and $J_{-2}$.
So far as I can see, if defining a $\mathbb{Z}$-graded ring from a sequence of abelian groups, we must have $J_k$ for all $k\in\mathbb{Z}$, and maps $J_k\times J_l\to J_{k+l}$ for all $k,l\in\mathbb{Z}$, where these maps satisfying various properties. Then $J:=\bigoplus_{k}J_k$ as an abelian group and we obtain a ring multiplication $J\times J\to J$ from these maps.
What am I missing?