nLab quasiregular element

Definition

In rings

Let $R$ be a ring. An element $r \in R$ is quasiregular if $1 - r$ is a unit. $r$ is left quasiregular if $1 - r$ is left invertible, and $r$ is right quasiregular if $1 - r$ is right invertible.

In rigs

Let $R$ be a rig. An element $r \in R$ is left quasiregular if one can construct an element $s \in R$ such that $s = s r + 1$. An element $r \in R$ is right quasiregular if one can construct an element $s \in R$ such that $s = r s + 1$. An element $r \in R$ is quasiregular if it is both left and right quasiregular. If $R$ is a ring, one can show that these conditions is equivalent to $1 - r$ being a unit.

Properties

Theorem

Suppose that addition in a rig $R$ is cancellative. If the multiplicative neutral element $1 \in R$ is a quasiregular element $R$, then $R$ is the trivial rig.

Proof

Suppose $1 \in R$ is quasiregular. We can construct element $s \in R$ such that $s = s1 + 1$, thus $s = s + 1$. By the cancellative property, we have $1 = 0$, meaning that $R$ is the trivial ring.

Lemma

The only ring where $1 \in R$ is quasiregular is the trivial ring.

Theorem

Every nilpotent element in a ring is a quasiregular element.

Proof

Given a ring $R$ and an element $x \in R$, if $x$ is nilpotent, then one can construct a natural number $n \in \mathbb{N}$ such that $x^{n + 1} = 0$, and

$$(1 - x)(1 + x + x^2 + \ldots + x^n) = 1 \qquad (1 + x + x^2 + \ldots + x^n)(1 - x) = 1$$

which means that $x$ is quasiregular.

Related concepts

• quasiregular rig

• Jacobson radical

References

• Wikipedia, Quasiregular element