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1
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Positive characteristic analogs of closed polynomials

100%
Jędrzejewicz P.
Open Mathematics
|
2011
|
tom 9
|
nr 1
50-56
EN
The notion of a closed polynomial over a field of zero characteristic was introduced by Nowicki and Nagata. In this paper we discuss possible ways to define an analog of this notion over fields of positive characteristic. We are mostly interested in conditions of maximality of the algebra generated by a polynomial in a respective family of rings. We also present a modification of the condition of integral closure and discuss a condition involving partial derivatives.
2
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A characterization of p-bases of rings of constants

100%
Jędrzejewicz P.
Open Mathematics
|
2013
|
tom 11
|
nr 5
900-909
EN
We obtain two equivalent conditions for m polynomials in n variables to form a p-basis of a ring of constants of some polynomial K-derivation, where K is a unique factorization domain of characteristic p > 0. One of these conditions involves Jacobians while the other some properties of factors. In the case m = n this extends the known theorem of Nousiainen, and we obtain a new formulation of the Jacobian conjecture in positive characteristic.
3
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On derivations of quantales

88%
Xiao Q., Liu W.
Open Mathematics
|
2016
|
tom 14
|
nr 1
338-346
EN
A quantale is a complete lattice equipped with an associative binary multiplication distributing over arbitrary joins. We define the notions of right (left, two) sided derivation and idempotent derivation and investigate the properties of them. It’s well known that quantic nucleus and quantic conucleus play important roles in a quantale. In this paper, the relationships between derivation and quantic nucleus (conucleus) are studied via introducing the concept of pre-derivation.
4
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Integrations on rings

88%
Banič I.
Open Mathematics
|
2017
|
tom 15
|
nr 1
365-373
EN
In calculus, an indefinite integral of a function f is a differentiable function F whose derivative is equal to f. The main goal of the paper is to generalize this notion of the indefinite integral from the ring of real functions to any ring. We also investigate basic properties of such generalized integrals and compare them to the well-known properties of indefinite integrals of real functions.
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