# Factoids #0

2019-06-15, post № 216

**mathematics**, #algebra, #rings

## I) Unit polynomials with non-vanishing degree

is its own multiplicative inverse, showing that does not hold in a general commutative Ring with one.

This phenomenon is uniquely characterized by the following equivalence:

Existence of the desired ring elements is assured by the following construction.

- Let rise discretely.
- If , implying , holds, since the construction arrived at this point, one finds
- The above condition is met for at least one , since otherwise would imply , which is impossible since and is a unit element.

As a corollary, the property follows for any integral domain.

Furthermore, looking at , this ring’s zero divisors are , with no mutual zero divisors summing to zero. Using the above, 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:

*Thanks to Nathan Tiggemann for finding this marvelous algebraic structure.*

Generalizing, any commutative ring with one 𝑅 induces a non-commutative ring without one on which 𝛬 acts as an omni-right-annihilator, namely

As a corollary, by constructing the above ring using the reals, one obtains a ring with a (left-factored) polynomial ring housing a polynomial of degree one having uncountably many roots: