# Factoids #0

2019-06-15, post № 216

mathematics, #algebra, #rings

## I) Unit polynomials with non-vanishing degree

$2t+1\in\mathbb{Z}_4[t]$ is its own multiplicative inverse, showing that $R[t]^*=R^*$ does not hold in a general commutative Ring with one.

This phenomenon is uniquely characterized by the following equivalence:

Proof. Negated replication. Let $R\not\owns f=\sum_{i=0}^n\alpha_it^i\in R[t]^*,\alpha_n\neq 0$ be a unit polynomial of non-vanishing degree $n\geq 1$. Let $g=\sum_{j=0}^m\beta_jt^j\in R[t]^*,\beta_m\neq 0$ denote its multiplicative inverse, i. e. $f\cdot g=1$.
Claim. The polynomial $g$ has non-vanishing degree $m\geq 1$.
Proof. Suppose $g\in R$. Since $f\cdot g=\sum_{i=0}^n(\alpha_i\cdot g)t^i$, it follows from $\alpha_n\cdot g=0$ that $g$ is a zero divisor. However, at the same time $a_0\cdot g=1$ implies that $g$ is a unit, arriving at a contradiction.
Since both $n,m\geq 1$, one concludes $\exists 1\leq k\leq m$ as well as $\alpha_n\cdot\beta_m=0$.
Existence of the desired ring elements $a,b$ is assured by the following construction.
• Let $k=1\nearrow m$ rise discretely.
• If $a:=\alpha_n\beta_{m-k}\neq 0$, implying $b:=\sum_{i=1}^k\alpha_{n-i}\beta_{m-k+i}\neq 0$, holds, since the construction arrived at this point, one finds
• The above condition is met for at least one $1\leq k\leq m$, since otherwise $k=m$ would imply $\alpha_n\beta_{m-m}=0$, which is impossible since $\alpha_n\neq 0$ and $\beta_0$ is a unit element.
By construction, $0\neq a,b$ as well as $a+b=0$ are given.
Negated implication. Setting $f:=at+1$, $g:=bt+1$, one calculates
showing $R\not\owns f,g\in R[t]^*$.

As a corollary, the property $R[t]^*=R^*$ follows for any integral domain.

Furthermore, looking at $\mathbb{Z}/6\mathbb{Z}$, this ring’s zero divisors are $\{0,2,3,4\}$, with no mutual zero divisors summing to zero. Using the above, $\mathbb{Z}/6\mathbb{Z}[t]^*=\mathbb{Z}/6\mathbb{Z}^*$ follows.

## II) A closing bijection

It defines

an isomorphism in the category Set.

## III) A ring full of zero divisors

It defines

a non-commutative ring without one of cardinality four in which every element is a zero divisor with left-annihilating element 𝛬: