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.
Setting , one calculates
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
an isomorphism in the category Set.
III) A ring full of zero divisors
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: