Is there a theory of "rings" with partially defined multiplication?

Question

Consider the abelian group $R [[\mathbb{Z}^d]]$ of all formal Laurent series over a commutative ring $R$ (a typical element has the form $\sum_{v \in \mathbb{Z}^d} \lambda_v \cdot X_1^{v_1} \dots X_d^{v_d}$ with $\lambda_v \in R$). Since there is no restriction on the coefficients, we cannot make $R [[\mathbb{Z}^d]]$ into a ring by the usual multiplication, but we get a partially defined operation in this way. It is easy to see that $R [[\mathbb{Z}^d]]$ satisfies all ring axioms, whenever all involved terms are defined. Maybe one could call such structures "partial rings".

Is there a general theory of partial (commutative) rings which abstracts from this example? I am particularly interested in the notion of ideals and quotients of them.

Thank you in advance!

Answer 1

Groupoids in the category theory can be described as partial groups. What you are asking for is called ringoids, see also here.

Answer 2

I don't have a proper answer, but here's an idea:

There's a category $\mathbf{Ab}_{\mathrm{par}}$ of abelian groups and partial homomorphisms between them. You should be able to make $\mathbf{Ab}_{\mathrm{par}}$ into a symmetric monoidal category with the usual tensor product. Then define that a partial ring is a monoid object in $\mathbf{Ab}_{\mathrm{par}}.$ Maybe try searching for the phrase "monoid object in the the monoidal category of abelian groups with partial homomorphisms" or something like that.