> For the complete documentation index, see [llms.txt](https://adams-personal-organization.gitbook.io/compendium-of-predicates/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://adams-personal-organization.gitbook.io/compendium-of-predicates/checks/algebra/boolean-ring.md).

# Boolean Ring

### Has Zero Divisors

<details>

<summary><span class="math">\exists(p,q \in A - \text{Zero}: p \neq q : p \otimes q = \text{Zero})</span></summary>

***

$$\textbf{BooleanRing} ; A ; \otimes ; \oplus ; \text{Zero} ; \text{One}$$

$$#A > 2$$

***

```
sig A {
  , tms: A->A
  , pls: A->A
}

sig Zero, One in A {}

has_zero_divisors: check {
  { BooleanRing[A,tms,pls,Zero,One]
    #A > 2
  } implies {
    some disj p,q: A-Zero {
      tms[p,q] = Zero
    }
  }
} for 10
```

</details>