# nLab multiplicatively cancellable rig

Definition

## Definition

In classical mathematics, a multiplicatively cancellable rig is a rig $R$ such that each $(r \neq 0) \in R$ is multiplicatively cancellable.

An element $r \in R$ where $R$ is a rig is left multiplicatively cancellable iff for every $s,t \in R$ we have that $(r.s = r.t) \Rightarrow (s = t)$ and it is right multiplicatively cancellable iff for every $s,t \in R$ we have that $(s.r = t.r) \Rightarrow (s = t)$. An element $r \in R$ is multiplicatively cancellable iff it is left multiplicatively cancellable and right multiplicatively cancellable.

If $R$ is a commutative rig, an element is left multiplicatively cancellable iff it is right multiplicatively cancellable.

### In constructive mathematics

Like in the case for integral domains and fields, the definition above bifurcates into multiple definitions in constructive mathematics.

definitions to be ported over

###### Definition

A residue multiplicatively cancellable rig is a rig $R$ such that for all $r \in R$, if $r$ is not multiplicatively cancellable, then $r = 0$.

###### Definition

A Heyting multiplicatively cancellable rig is a rig $R$ with a tight apartness relation $\#$ such that each $(r \# 0) \in R$ is multiplicatively cancellable.

###### Definition

A discrete multiplicatively cancellable rig is a rig $R$ such that each $r \in R$ is zero xor multiplicatively cancellable.

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