# Interpretation of preadditive categories as rings.

My motivation comes from How to see that a one-object pre-additive category is a ring? and this master thesis. Basically, I want to see concrete examples of preadditive categories and what structure they induce.

The basic example

A preadditive category $\mathcal{A}$ with $obj(\mathcal{A}) = \{X\}$ induces a ring structure in the sets of morphisms $(\mathcal{A}(X,X),+,\circ)$.

• This question is rather unclear. In particular when you're talking about the induced structure it isn't clear on what object are you actually inducing a structure. Your first example is either wrong or contains a typo: if $\mathcal A$ and $X=\mathcal A$ it doesn't make any sense the data $\mathcal A(X,X)=\mathcal A(\mathcal A,\mathcal A)$. Could you add some details and try to be a little more specific? Otherwise it is very difficult to address your problem. – Giorgio Mossa May 1 '18 at 15:22
• Incidentally, another example that seems to be missing from the source material is that if Z is an abelian group, then a Z-graded ring $A$ can be expressed as a preadditive category whose objects are the elements of Z and $\hom(z, z')$ is the subgroup of $A$ of elements of degree $z' - z$, with composition given by the product in $A$. This can furthermore be made into a monoidal category, with the monoidal operation given on objects by addition in $Z$ and on morphisms by the product in $A$. – Hurkyl May 18 '18 at 15:12