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:
Proof. Negated replication.
Let be a unit polynomial of non-vanishing degree
. Let
denote its multiplicative inverse, i.e.
.
Claim. The polynomial has non-vanishing degree
.
Proof. Suppose . Since
, it follows from
that
is a zero divisor. However, at the same time
implies that
is a unit, arriving at a contradiction.
Since both , one concludes
as well as
.
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.
By construction, as well as
are given.
Negated implication.
Setting , one calculates
,
showing .
q.e.d.
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:
.