Some Algebraic Properties of Polynomial Rings

In this article we extend the algebraic theory of polynomial rings, formalized in Mizar [1], based on [2], [3]. After introducing constant and monic polynomials we present the canonical embedding of R into R[X] and deal with both unit and irreducible elements. We also define polynomial GCDs and show that for fields F and irreducible polynomials p the field F[X]/ is isomorphic to the field of polynomials with degree smaller than the one of p.Schwarzweller Christoph - Institute of Computer Science University of Gdansk, PolandKorniłowicz Artur - Institute of Informatics University of Białystok, PolandRowinska-Schwarzweller Agnieszka - Sopot PolandGrzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1.H. Heuser. Lehrbuch der Analysis. B.G. Teubner Stuttgart, 1990.Steven H. Weintraub. Galois Theory. Springer Verlag, 2 edition, 2009

Monogenous algebras. Back to Kronecker

In this note we develop some properties of those algebras (called here locally simple) which can be generated by a single element after, if need be, a faithfully flat extension. For finite algebras, this is shown to be in fact a property of the geometric fibers. Morphisms between rings of algebraic integers are locally simple. Expanding an idea introduced by Kronecker we show that much of the properties (in particular local simplicity) of a finite and locally free A-algebra B can be read through the characteristic polynomial of the generic element of B.Comment: Note ecrite en septembre 2003. 13 page

On posets, monomial ideals, Gorenstein ideals and their combinatorics

In this article we first compare the set of elements in the socle of an ideal of a polynomial algebra $K[x_1,\ldots,x_d]$ over a field $K$ that are not in the ideal itself and Macaulay's inverse systems of such polynomial algebras in a purely combinatorial way for monomial ideals, and then develop some closure operational properties for the related poset {{\nats}_0^d}. We then derive some algebraic propositions of $\Gamma$-graded rings that then have some combinatorial consequences. Interestingly, some of the results from this part that uniformly hold for polynomial rings are always false when the ring is local. We finally delve into some direct computations, w.r.t.~a given term order of the monomials, for general zero-dimensional Gorenstein ideals and deduce a few explicit observations and results for the inverse systems from some recent results about socles.Comment: 36 pages, a few new examples added, a reference added, some rephrasing of confusing sentences were made and typos fixe