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Let's look at this set ($e_{i}=(\delta_{ij})_{j}$ for all $i,j\geq0$ (kronecker delta))

$R_{\mathbb{Z}} =$$\{\sum_{i=0}a_{i}e_{i}$ : $a_{i}\in \mathbb{Z}$ $ \land$ $ \exists k,A_1,A_2,A_3,...,A_k\in\mathbb{Z}$ $\forall n\geq k$ $a_n=\sum_{i=1}^kA_ia_{n-i}\}$

We are able to say $R_{\mathbb{Z}} \subset\Pi \mathbb{Z}$. But i also predict $R_{\mathbb{Z}}$ is subring of $\Pi \mathbb{Z}$ because of this theorem:

$a_n=\sum_{i=1}^kA_ia_{n-i}\iff \sum_{i=0}a_ix^i=\dfrac {(\sum_{n=1}^{k-1}A_nx^n\sum_{i=0}^{k-n-1}a_ix^i)-\sum_{i=0}^{k-1}a_ix^i} {\sum_{i=1}^kA_ix^i-1}$

We can prove closure property with this. I couldn't prove closure property of multiplication. How can i do that? And is there an article about this ring ?

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One can see from this article, that $R_\mathbb{Z}$ is a subring of $\prod_{n \in \mathbb{N}_0} \mathbb{Z}$.

I know no article which is explicitely about this ring and I do not think that there is one. But note that your ring contains all constant recurrences $(a)_{n \in \mathbb{N}_0}$ and therefore is a ring of functions in the terminology of On the spectrum of rings of functions by Frisch (you can find it freely available on her homepage). It can also easily seen to be divisible (which is also defined in the same paper) and therefore Corollary 6.5 applies which gives nice characterizations of maximal ideals and residue rings of your ring.

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