Some Results on Noetherian Warfield Domains

Let [Formula: see text] be a domain. In this paper, we show that if [Formula: see text] is one-dimensional, then [Formula: see text] is a Noetherian Warfield domain if and only if every maximal ideal of [Formula: see text] is 2-generated and for every maximal ideal[Formula: see text] of [Formula: see text], [Formula: see text] is divisorial in the ring [Formula: see text]. We also prove that a Noetherian domain [Formula: see text] is a Noetherian Warfield domain if and only if for every maximal ideal [Formula: see text] of [Formula: see text], [Formula: see text] can be generated by two elements. Finally, we give a sufficient condition under which all ideals of [Formula: see text] are strongly Gorenstein projective.

$$v_c$$-Noetherian domains and Krull domains

Let D be an integrally closed domain, $$\{V_{\alpha }\}$$ { V α } be the set of t-linked valuation overrings of D, and $$v_c$$ v c be the star operation on D defined by $$I^{v_c} = \bigcap _{\alpha } IV_{\alpha }$$ I v c = ⋂ α I V α for all nonzero fractional ideals I of D. In this paper, among other things, we prove that D is a $$v_c$$ v c -Noetherian domain if and only if D is a Krull domain, if and only if $$v_c = v$$ v c = v and every prime t-ideal of D is a maximal t-ideal. As a corollary, we have that if D is one-dimensional, then $$v_c = v$$ v c = v if and only if D is a Dedekind domain.

Homogeneous localizations of some quantum enveloping superalgebras

Under suitable conditions the skewfield of fractions of a superalgebra which is a noetherian domain is canonically provided with a structure of superalgebra. This gives rise to a notion of rational equivalence in the category of superalgebras. We study from the point of view of this rational equivalence some low dimensional examples of quantum enveloping algebras of Lie superalgebras.

Degree of Rational Maps and Specialization

One considers the behavior of the degree of a rational map under specialization of the coefficients of the defining linear system. The method rests on the classical idea of Kronecker as applied to the context of projective schemes and their specializations. For the theory to work, one is led to develop the details of rational maps and their graphs when the ground ring of coefficients is a Noetherian domain.

Noetherian domains in which every ideal is isomorphic to a trace ideal

This paper seeks an answer to the following question: Let [Formula: see text] be a Noetherian ring with [Formula: see text]. When is every ideal isomorphic to a trace ideal? We prove that for a local Noetherian domain [Formula: see text] with [Formula: see text], every ideal is isomorphic to a trace ideal if and only if either [Formula: see text] is a DVR or [Formula: see text] is one-dimensional divisorial domain, [Formula: see text] is a principal ideal of [Formula: see text] and [Formula: see text] posses the property that every ideal of [Formula: see text] is isomorphic to a trace ideal of [Formula: see text]. Next, we globalize our result by showing that a Noetherian domain [Formula: see text] with [Formula: see text] has every ideal isomorphic to a trace ideal if and only if either [Formula: see text] is a PID or [Formula: see text] is one-dimensional divisorial domain, every invertible ideal of [Formula: see text] is principal and for every non-invertible maximal ideal [Formula: see text] of [Formula: see text], [Formula: see text] is a principal ideal of [Formula: see text] and every ideal of [Formula: see text] is isomorphic to a trace ideal of [Formula: see text]. We close the paper by examining some classes of non-Noetherian domains with this property to provide a large family of original examples.

On 1-absorbing primary ideals of commutative rings

Let [Formula: see text] be a commutative ring with nonzero identity. In this paper, we introduce the concept of 1-absorbing primary ideals in commutative rings. A proper ideal [Formula: see text] of [Formula: see text] is called a [Formula: see text]-absorbing primary ideal of [Formula: see text] if whenever nonunit elements [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] Some properties of 1-absorbing primary ideals are investigated. For example, we show that if [Formula: see text] admits a 1-absorbing primary ideal that is not a primary ideal, then [Formula: see text] is a quasilocal ring. We give an example of a 1-absorbing primary ideal of [Formula: see text] that is not a primary ideal of [Formula: see text]. We show that if [Formula: see text] is a Noetherian domain, then [Formula: see text] is a Dedekind domain if and only if every nonzero proper 1-absorbing primary ideal of [Formula: see text] is of the form [Formula: see text] for some nonzero prime ideal [Formula: see text] of [Formula: see text] and a positive integer [Formula: see text]. We show that a proper ideal [Formula: see text] of [Formula: see text] is a 1-absorbing primary ideal of [Formula: see text] if and only if whenever [Formula: see text] for some proper ideals [Formula: see text] of [Formula: see text], then [Formula: see text] or [Formula: see text]

On S-GCD domains

Let [Formula: see text] be a multiplicative set in an integral domain [Formula: see text]. A nonzero ideal [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-[Formula: see text]-principal if there exist an [Formula: see text] and [Formula: see text] such that [Formula: see text]. Call [Formula: see text] an [Formula: see text]-GCD domain if each finitely generated nonzero ideal of [Formula: see text] is [Formula: see text]-[Formula: see text]-principal. This notion was introduced in [A. Hamed and S. Hizem, On the class group and [Formula: see text]-class group of formal power series rings, J. Pure Appl. Algebra 221 (2017) 2869–2879]. One aim of this paper is to characterize [Formula: see text]-GCD domains, giving several equivalent conditions and showing that if [Formula: see text] is an [Formula: see text]-GCD domain then [Formula: see text] is a GCD domain but not conversely. Also we prove that if [Formula: see text] is an [Formula: see text]-GCD [Formula: see text]-Noetherian domain such that every prime [Formula: see text]-ideal disjoint from [Formula: see text] is a [Formula: see text]-ideal, then [Formula: see text] is [Formula: see text]-factorial and we give an example of an [Formula: see text]-GCD [Formula: see text]-Noetherian domain which is not [Formula: see text]-factorial. We also consider polynomial and power series extensions of [Formula: see text]-GCD domains. We call [Formula: see text] a sublocally [Formula: see text]-GCD domain if [Formula: see text] is a [Formula: see text]-GCD domain for every non-unit [Formula: see text] and show, among other things, that a non-quasilocal sublocally [Formula: see text]-GCD domain is a generalized GCD domain (i.e. for all [Formula: see text] is invertible).

A short note on annihilators of local cohomology modules

Let [Formula: see text] be a commutative Noetherian domain, [Formula: see text] a nonzero [Formula: see text]-module of finite injective dimension, and [Formula: see text] be a nonzero ideal of [Formula: see text]. In this paper, we prove that whenever [Formula: see text], then the annihilator of [Formula: see text] is zero. Also, we calculate the annihilator of [Formula: see text] for finitely generated [Formula: see text]-modules [Formula: see text] and [Formula: see text] with conditions [Formula: see text] and [Formula: see text]. Moreover, if [Formula: see text] is a regular Noetherian local ring and [Formula: see text] such that [Formula: see text], then we show that there exists an ideal [Formula: see text] of [Formula: see text] such that [Formula: see text], [Formula: see text] and [Formula: see text].

ON SEPARABLE AND -FORMS

In this paper, we will prove that any $\mathbb{A}^{3}$-form over a field $k$ of characteristic zero is trivial provided it has a locally nilpotent derivation satisfying certain properties. We will also show that the result of Kambayashi on the triviality of separable $\mathbb{A}^{2}$-forms over a field $k$ extends to $\mathbb{A}^{2}$-forms over any one-dimensional Noetherian domain containing $\mathbb{Q}$.