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Preprint Number 753
753. Leigh Evron, Joseph R. Mileti, and Ethan Ratliff-Crain Irreducibles and primes in computable integral domains E-mail: Submission date: 22 July 2014. Abstract: A computable ring is a ring equipped with mechanical procedure to add and
multiply elements. In most natural computable integral domains, there is a
computational procedure to determine if a given element is prime/irreducible.
However, there do exist computable UFDs (in fact, polynomial rings over
computable fields) where the set of prime/irreducible elements is not
computable. Outside of the class of UFDs, the notions of irreducible and prime
may not coincide. We demonstrate how different these concepts can be by
constructing computable integral domains where the set of irreducible elements
is computable while the set of prime elements is not, and vice versa. Along the
way, we will generalize Kronecker's method for computing irreducibles and
factorizations in Mathematics Subject Classification: Keywords and phrases: |

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