# Emma Nestor 🍏

Posted .

(Solution to the challenge at the end of Doing basic group theory with diagrams)

Given a set $$R$$, a singleton set $$I$$, and functions

• $$0:I\to R$$ (additive identity),
• $$\mathrm{add}:R\times R\to R$$ (addition),
• $$\mathrm{neg}:R\to R$$ (negative),
• $$1:I\to R$$ (multiplicative identity),
• $$\mathrm{mul}:R\times R\to R$$ (multiply).

Then $$(R,\mathrm{add},0)$$ needs to be a group object, which is abelian (using the swap function), and $$(R,\mathrm{mul},1)$$ needs to satisfy associativity and identity. In order to make it a ring object we need multiplication to distribute over addition, i.e. the following diagrams commute:

• Left-distributivity:
• Right-distributivity: