# Generalized Formal Power Series Ring

I came up with the following generalization of formal power series and wonder if anyone knows a reference where this is studied.

Let $R$ be a commutative ring with unity and $M$ a commutative monoid written additively. Consider a family of subsets $\mathcal{T} \subseteq \mathcal{P}(M)$ with the following properties:

1. There is $I \in \mathcal{T}$ with $0 \in I$.
2. For $I, J \in \mathcal{T}$ there is $K \in \mathcal{T}$ with $I \cup J \subseteq K$.
3. For $I, J \in \mathcal{T}$ there is $K \in \mathcal{T}$ with $I + J \subseteq K$.
4. For all $I, J \in \mathcal{T}, m \in M$ the set $\{(i, j) \in I \times J \mid i + j = m\}$ is finite.
5. If $I \in \mathcal{T}$ then for all $J \subseteq I$ we have $J \in \mathcal{T}$.
6. If $m \in M$ then there is $I \in \mathcal{T}$ with $m \in I$.

Define $R[[M]]_\mathcal{T}$ to be the set of functions $f : M \to R$ such that there is $I \in \mathcal{T}$ with $\operatorname{supp}(f) \subseteq I$. We write elements in $R[[M]]_\mathcal{T}$ as formal sums $$\sum_{m \in M} a_m X^m.$$ Moreover we define addition point-wise and multiplication by convolution, i.e. $$(\sum_{m \in M} a_m X^m)(\sum_{m \in M} b_m X^m) = \sum_{m \in M} (\sum_{i+j = m} a_i b_j) X^m.$$ I shortly discuss the properties above.

1. We have an embedding $R \to R[[M]]_\mathcal{T}$.

3+4. The convolution product is well-defined.

5+6. These are just properties to make everything a bit easier. I.e. $\operatorname{supp}(f) \in \mathcal{T}$, the union and sum of two sets is in $\mathcal{T}$. If $\mathcal{T}$ does not satisfy 5 or 6 we can consider $$\widetilde{\mathcal{T}} := \{I \subseteq J \mid J \in \mathcal{T}\} \text{ and } \widetilde{M} := \{m \in I \mid I \in \mathcal{T}\}$$ which then have the properties 1-6. In this case, in 2 and 3 equality can be assumed.

The usual monoid rings are obtained by considering the family of finite sets. The ring of formal power series $R[[X]]$ is obtained by considering $M = \mathbb{N}$ and $\mathcal{T} = \mathcal{P}(\mathbb{N})$. The ring of formal laurent series $R((X)$ is obtainable by taking $M = \mathbb{Z}$ and the family of sets which are bounded from below.

So the questions are:

1. Are there any references where this is studied

2. Universal Property?

3. What properties from $R$ are transferred to $R[[M]]_\mathcal{T}$ under which assumptions on $M$ and $\mathcal{T}$?

4. Is there some "natural" $\mathcal{T}'$ on the Grothendieck group and how do the corresponding rings relate? What about products?

5. Analogous questions about the usual constructions of rings, i.e. quotients, products, localizations, ...

I'm relatively sure that this is in generally not easy to answer, since already the group rings $R[G]$ have a lot of unanswered questions about being integral domains and their unit groups. Questions 4 and 5 should be answerable, at least for product rings and products of monoids I'm relatively sure.

• Is there a universal property for the "usual" ring of power series $R[[X]]$ ? – Max May 23 '17 at 20:29
• – Paul K May 23 '17 at 20:30

The following does not anwer your questions but it is too long to be a comment:

There is a notion that is less general than yours in the context of ordered monoïds/groups: Hahn series fields.

If $M$ is an ordered monoïd, then the set $\mathcal{T}_{or}$ of well-ordered subsets of $M$ satisfies 1-6. What's more, if $R$ is actually a field and $M$ is actually an ordered group, $R[[M]]_{\mathcal{T}_{or}}$ is a field, usually denoted $R((x^M))$. Provided $R$ is ordered, those structures can be ordered by rendering every series with a positive first coefficient positive.

Note that this covers all the examples you gave when equipped with usual orders. The assumption that the monoïd/group is ordered does not come from a focus on ordered structures but rather naturally appears when studying some fundamental objects of valuation-theory which in turn appear for instance in algebraic geometry and field theory.

In the case of Hahn series fields, they are universal for valued fields of given residue fields and value groups under some conditions (see Maximal fields with valuation of Irving Kaplansky for more precisions), but do not satisfy a universal property known to me.

For Hahn series fields if $F((x^G))$ is algebraically closed iff $F$ is and $G$ is divisible. There are other nice transfers of properties.

• Thanks for dropping the name "Hahn series", looked it up now. Do you know some good references? – Paul K May 23 '17 at 21:14
• Not really. I have only seen properties 3) and 4) clearly proven by Bernhard Neumann in On ordered division rings (the original paper of Hahn is in German). He aslo proves the existence of inverses when the base ring is a field, and I think Hahn used another method which I guess you could understand if you know a little German. Kaplansky is a very good reference for valued fields. – nombre May 23 '17 at 21:37