A polynomial f (t) in an Ore extension K[t; S, D] over a division ring K is a Wedderburn polynomial if f (t) is monic and is the minimal polynomial of an algebraic subset of K. These polynomials have been studied in [LL 5 ]. In this paper, we continue this study and give some applications to triangulation, diagonalization and eigenvalues of matrices over a division ring in the general setting of (S, D)-pseudo-linear transformations. In the last section we introduce and study the notion of G-algebraic sets which, in particular, permits generalization of Wedderburn's theorem relative to factorization of central polynomials.dim C ker λ f,a = dim C ker λ h,a + dim C (Im λ h,a ∩ ker λ g,a ). But this is exactly what is given by Lemma 3.3.
The information overload problem, also known as infobesity, forces online vendors to utilize collaborative filtering algorithms. Although various recommendation methods are widely used by many electronic commerce sites, they still have substantial problems, including but not limited to privacy, accuracy, online performance, scalability, cold start, coverage, grey sheep, robustness, being subject to shilling attacks, diversity, data sparsity, and synonymy. Privacy-preserving collaborative filtering methods have been proposed to handle the privacy problem. Due to the increasing popularity of privacy protection and recommendation estimation over the Internet, prediction schemes with privacy are still receiving increasing attention. Because research trends might change over time, it is critical for researchers to observe future trends. In this study, we determine the existing trends in the privacypreserving collaborative filtering field by examining the related papers published mainly in the last few years. Comprehensive examinations of the most up-to-date related studies are described. By scrutinizing the contemporary inclinations, we present the most promising possible research trends in the near future. Our proposals can help interested researchers direct their research toward better outcomes and might open new ways to enrich privacy-preserving collaborative filtering studies.
Let R denote a 2-fir. The notions of F -independence and algebraic subsets of R are defined. The decomposition of an algebraic subset into similarity classes gives a simple way of translating the F -independence in terms of dimension of some vector spaces. In particular to each element a ∈ R is attached a certain algebraic set of atoms and the above decomposition gives a lower bound of the length of the atomic decompositions of a in terms of dimensions of certain vector spaces. A notion of rank is introduced and fully reducible elements are studied in details.Proof. Of course, we will only prove that i) implies ii). So let s, t ∈ R be such that Rs ∩ Rt = 0. We can find a, b ∈ R such that 0 = as = bt and i) shows that there exist c, d ∈ R such that aR ∩ bR = cR and aR + bR = dR. Writing c = ab ′ = ba ′ , a = dx, and b = dy, we get dxb ′ = ab ′ = ba ′ = dya ′ . Since R is a domain this gives xb ′ = ya ′ and we easily obtain that xR∩yR = xb ′ R = ya ′ R and xR + yR = R. Lemma 1.1 shows that Ra ′ + Rb ′ = R and Ra ′ ∩ Rb ′ = Rxb ′ = Rya ′ . Now, since as = bt ∈ aR ∩ bR = cR there exists v ∈ R such that as = bt = cv = ab ′ v = ba ′ v and so s = b ′ v and t = a ′ v. We thus get the desired conclusions : Rs + Rt = Rv and Rs ∩ Rt = Ru for u = xb ′ v.Lemma 1.1 and Corollary 1.2 will be used several times. For more details on 2-fir we refer to P.M. Cohn's book "Free rings and their relations" ([3]). We
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