#### 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:

.