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: