Semistar ascending chain conditions over polynomial rings

Ricerche di Matematica - Let D be an integral domain. G. Picozza associated to a stable semistar operation $$\star $$ on D,  a semistar operation $$\star _1$$ on the polynomial ring D[X]....     

A note on nil power serieswise Armendariz rings

A ring R is called nil power serieswise Armendariz if $ \forall f = \sum\limits_{i = 0}^\infty {a_i X^i } $ \forall f = \sum\limits_{i = 0}^\infty {a_i X^i } and $ g = \sum\limits_{i = 0}^\infty {b_i X^i } $ g = \sum\limits_{i = 0}^\infty {b_i X^i } in R[[X]] such that f g ∈ Nil(R)[[X]], then a _i b _j ∈ Nil(R) for all i and j. In this note we characterize completely nil power serieswise Armendariz rings with their nilradical Nil(R) (where the nilradical is the set of nilpotent elements). We prove that a ring is nil power serieswise Armendariz if and only if Nil(R) is an ideal of R. We prove that each power serieswise Armendariz ring is nil power serieswise Armendariz and we give examples of nil power serieswise Armendariz rings.     

Convergence to the equilibrium for the Pauli equation without detailed balance condition

Let $A?B$ be an extension of commutative rings with identity, X an analytic indeterminate over B, and $R\text{:}=A+XB[[X]]$, the subring of the formal power series ring $B[[X]]$, consisting of the series with constant terms in A. In this Note we study when the ring R is Noetherian. We prove that R is Noetherian if and only if A is Noetherian and B is a finitely generated A-module. To cite this article: S. Hizem, A. Benhissi, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On the class group and S-class group of formal power series rings

Let ${\text{Cl}}_t(A)$ denote the t-class group of an integral domain A. P. Samuel has established that if A is a Krull domain then the mapping ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, is injective and if A is a regular UFD, then ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, is bijective. Later, L. Claborn extended this result in case A is a regular Noetherian domain. In the first part of this paper we prove that the mapping ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$; $[I]?[{(I.A?X?)}_t]$ is an injective homomorphism and in case of an integral domain A such that each υ-invertible υ-ideal of A has υ-finite type, we give an equivalent condition for ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, to be bijective, thus generalizing the result of Claborn. In the second part of this paper, we define the S-class group of an integral domain A: let S be a (not necessarily saturated) multiplicative subset of an integral domain A. Following [11], a nonzero fractional ideal I of A is S-principal if there exist an $s\in S$ and $a\in I$ such that $sI?aA?I$. The S-class group of A, S-${\text{Cl}}_t(A)$ is the group of fractional t-invertible t-ideals of A under t-multiplication modulo its subgroup of S-principal t-invertible t-ideals of A. We generalize some known results developed for the classic contexts of Krull and PυMD domain and we investigate the case of isomorphism S-${\text{Cl}}_t(A)?S$-${\text{Cl}}_t(A?X?)$.     

Semistar ascending chain conditions over power series rings

Ricerche di Matematica - Let D be an integral domain. We associate to a semistar operation $$\star $$ on D, a semistar operation $$*$$ on D[[X]]. We prove that if D satisfies the $$\star _f$$...     

Modules Satisfying the S-Noetherian Property and S-ACCR

Let A = k[x₁,..., x_n] be a standard graded k-algebra over an infinite field k. We assume A is Cohen-Macaulay and has Krull dimension one. Let e denote the multiplicity of A and r - 1 the postulation number of A. Let I be a homogeneous ideal in A of grade one. Let j(I) denote the smallest degree of a regular form in I. Let l (I) denote the smallest power of (x₁,... , x_n) contained in I. If μ(I) denotes the minimum number of generators of I, then we show μ(I) ≤ e + j(I) - max{r,l(I)}, We then show how Dubreil's Second Theorem follows easily from this inequality     

Identification of the double and single acceptor state of isolated NiGa in GaAs

The question of the exact energy positions of isolated Ni_Ga as well as their optical and electrical properties have not yet been fully clarified in GaAs. We present a systematic study by deep-level transient spectroscopy, deep-level optical spectroscopy and optical absorption performed on several Ni-doped GaAs materials: n- and p-type LEC (liquid-encapsulated Czochralski) grown and p-type layers grown by liquid-phase epitaxy (LPE). All the electrical and optical results are up to now relatively coherent with the following identifications: (i) the double-acceptor charge state (Ni⁺ /Ni²⁺) is at E_c - 0.4 eV, (ii) the single-acceptor charge state (Ni²⁺ / Ni³⁺) is at E_v + 0.2 eV. However, when Ni is introduced during LPE in p-type materials we do not detect the Ni²⁺ /Ni³⁺ level which suggest a very low solubility of Ni in the LPE growth conditions.     

S-Noetherian Rings of the Forms 𝒜[X] and 𝒜[[X]]

Let 𝒜 = (A _n ) _n≥0 be an ascending chain of commutative rings with identity, S ? A ₀ a multiplicative set of A ₀, and let 𝒜[X] (respectively, 𝒜[[X]]) be the ring of polynomials (respectively, power series) with coefficient of degree i in A _i for each i ∈ ?. In this paper, we give necessary and sufficient conditions for the rings 𝒜[X] and 𝒜[[X]] to be S ? Noetherian.