MATRIKS ATAS RING DERET PANGKAT TERGENERALISASI MIRING

Let R be a ring with unit elements, strictly ordered monoids, and a monoid homomorphism. Formed , which is a set of all functions from S to R with are Artin and narrow. With the operation of the sum of functions and convolution multiplication, is a ring, from now on referred to as the Skew Generalized Power Series Ring (SGPSR). In this paper, the set of all matrices over SGPSR will be constructed. Furthermore, it will be shown that this set is a ring with the addition and multiplication matrix operations. Moreover, we will construct the ideal of ring matrix over SGPSR and investigate this ideal's properties.

On reduced archimedean skew power series rings

In this paper, we prove that if [Formula: see text] is an Archimedean reduced ring and satisfy ACC on annihilators, then [Formula: see text] is also an Archimedean reduced ring. More generally, we prove that if [Formula: see text] is a right Archimedean ring satisfying the ACC on annihilators and [Formula: see text] is a rigid automorphism of [Formula: see text], then the skew power series ring [Formula: see text] is right Archimedean reduced ring. We also provide some examples to justify the assumptions we made to obtain the required result.

Polynomial and power series ring extensions from sequences

Let [Formula: see text] be a commutative ring with identity. Let [Formula: see text] and [Formula: see text] be the collection of polynomials and, respectively, of power series with coefficients in [Formula: see text]. There are a lot of multiplications in [Formula: see text] and [Formula: see text] such that together with the usual addition, [Formula: see text] and [Formula: see text] become rings that contain [Formula: see text] as a subring. These multiplications are from a class of sequences [Formula: see text] of positive integers. The trivial case of [Formula: see text], i.e. [Formula: see text] for all [Formula: see text], gives the usual polynomial and power series ring. The case [Formula: see text] for all [Formula: see text] gives the well-known Hurwitz polynomial and Hurwitz power series ring. In this paper, we study divisibility properties of these polynomial and power series ring extensions for general sequences [Formula: see text] including UFDs and GCD-domains. We characterize when these polynomial and power series ring extensions are isomorphic to each other. The relation between them and the usual polynomial and power series ring is also presented.

Nonnil-Noetherian Rings and Formal Power Series

Let A be a commutative ring with unit. We characterize when A is nonnil-Noetherian in terms of the quotient ring A/ Nil(A) and in terms of the power series ring A[[X]].

On Nonnil-S-Noetherian Rings

Let R be a commutative ring with identity, and let S be a (not necessarily saturated) multiplicative subset of R. We define R to be a nonnil-S-Noetherian ring if each nonnil ideal of R is S-finite. In this paper, we study some properties of nonnil-S-Noetherian rings. More precisely, we investigate nonnil-S-Noetherian rings via the Cohen-type theorem, the flat extension, the faithfully flat extension, the polynomial ring extension, and the power series ring extension.

Higher Order Polars of Quasi-Ordinary Singularities

A quasi-ordinary polynomial is a monic polynomial with coefficients in the power series ring such that its discriminant equals a monomial up to unit. In this paper, we study higher derivatives of quasi-ordinary polynomials, also called higher order polars. We find factorizations of these polars. Our research in this paper goes in two directions. We generalize the results of Casas–Alvero and our previous results on higher order polars in the plane to irreducible quasi-ordinary polynomials. We also generalize the factorization of the first polar of a quasi-ordinary polynomial (not necessarily irreducible) given by the first-named author and González-Pérez to higher order polars. This is a new result even in the plane case. Our results remain true when we replace quasi-ordinary polynomials by quasi-ordinary power series.

The Sufficient Conditions for M[[S,w]] to be T[[S,w]]-Noetherian R[[S,w]]-module

In this paper, we investigate the sufficient conditions for T[[S,w]] to be a multiplicative subset of skew generalized power series ring R[[S,w]], where R is a ring, T Í R a multiplicative set, (S,≤) a strictly ordered monoid, and w : S®End(R) a monoid homomorphism. Furthermore, we obtain sufficient conditions for skew generalized power series module M[[S,w]] to be a T[[S,w]]-Noetherian R[[S,w]]-module, where M is an R-module.

A characterization of 2-primal and NI rings over skew inverse Laurent series rings

Let [Formula: see text] be an associative ring equipped with an automorphism [Formula: see text] and an [Formula: see text]-derivation [Formula: see text]. In this note, we characterize when a skew inverse Laurent series ring [Formula: see text] and a skew inverse power series ring [Formula: see text] are 2-primal, and we obtain partial characterizations for those to be NI.