It is known that the power series ring R[[x]] over a Dedekind finite ring R is Dedekind finite. Then the polynomial ring R[x] over R is also Dedekind finite as a subring of R[[x]]. But this procedure does not show the useful details that occur in the argument of the direct proof of Dedekind finiteness passing to polynomial rings. Thus, in this article, we provide R[x] with a direct proof, to study the structure of coefficients of the products of polynomials.
Let a be an ideal of a Noetherian ring R and M be a finitely generated R-module with cd(a, M) = c >= 0. Assume that the R-module H-a (c) (M) is Artinian and dim H-a(i) (M) <= 1 for all integers i >= 2. In this paper, it is shown that the R-module H-a (c) (M) is a-cofinite. Also, in this direction some of other results will be included.
. In this paper, we study a class of generalized Lambert W scope of transcendental equations that can be solved in mathematics, physics, and engineering. We analyze their basic properties, including real and complex branches, derivatives and integrals, Taylor series expansions, and branch structures. We also develop numerical schemes, such as Halley-type iterations, for higher-order generalizations. Practical applications are presented to illustrate the versatility of these functions, including quantile functions for Erlang and negative binomial distributions, delay differential equations, and physical models such as the double Dirac delta potential.
. In this paper, we study inclusion relations for a previously introduced class of harmonic close-to-convex functions. Using operators defined via the Pascal and Poisson distribution series and applying suitable coefficient bounds, we obtain new inclusion results among the classes of harmonic convex functions, harmonic starlike functions, and the defined class. Our findings highlight the effectiveness of distribution series operators in geometric function theory.
In this article, we will show some algebraic dependence theo-rems for meromorphic mappings into a projective space sharing few mov-ing hyperplanes with different multiplicities. Our result is an improve-ment of many previous results in this topic.
