Divisibility and groups in one-generated semirings

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F18%3A00322037" target="_blank" >RIV/68407700:21230/18:00322037 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1142/S0219498818500718" target="_blank" >http://dx.doi.org/10.1142/S0219498818500718</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1142/S0219498818500718" target="_blank" >10.1142/S0219498818500718</a>

Result language
angličtina
Original language name
Divisibility and groups in one-generated semirings
Original language description
Let (S,+, .) be a semiring generated by one element. Let us denote this element by w is an element of S and let g(x) is an element of x . N[x] be a polynomial. It has been proved that if g(x) contains at least two different monomials, then the elements of the form g(w) may possibly be contained in any countable commutative semigroup. In particular, divisibility of such elements does not imply their torsion. Let, on the other hand, g(x) consist of a single monomial (i.e. g(x) = kx(n), where k, n is an element of N). We show that in this case, the divisibility of g(w) by infinitely many primes implies that g(w) generates a group within (S, +). Further, an element a is an element of S is called an m-fraction of an element z is an element of S if m is an element of N and z = m . a. We prove that "almost every" m-fraction of w(n) can be expressed as f(w) for some polynomial f is an element of x . N[x] of degree at most n.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2018
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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