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
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: