Factoids #0 (#216)
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
- 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.
Negated implication. Setting
-=-
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: