The Quotient Ring of Algebraic Interger Ring

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Prüfer Conditions and the Total Quotient Ring

In this article, we consider two of the five well-studied extensions of the Prüfer domain notion to arbitrary commutative rings. In particular, we consider a class of rings that lies properly between Gaussian and Prüfer rings. We give a characterization of this property in terms of the total quotient ring and show that this property behaves nicely in conductor squares.     

The Quotient Category of a Graded Morita-Takeuchi Context

In this paper, we offer a graded equivalence between the quotient categories defined by any graded Morita-Takeuchi context via certain modifications of the graded cotensor functors. As a consequence, we show a commutative diagram that establish the relation between the closed objects of the categories gr^c and M^C, where C is a graded coalgebra.     

The Meta-Grothendieck Group of a Ring

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A formally Kähler structure on a knot space of a G 2-manifold

A knot space in a manifold M is a space of oriented immersions ${S^{1} \hookrightarrow M}$ up to Diff(S ¹). J.-L. Brylinski has shown that a knot space of a Riemannian threefold is formally Kähler. We prove that a space of knots in a holonomy G ₂ manifold is formally Kähler.     

ℤ2-Codimensions of the Grassmann Algebra

Let E be the infinite-dimensional Grassmann algebra over a field F of characteristic zero, and consider L the F-vector space spanned by all generators of E. Let ? _l be any fixed automorphism of E of order 2 such that L is an homogeneous subspace.

Our goal is to finish the computation of the sequences of ? _l -codimensions, by finding its exact value for the unique open case, that is, when the subspace of L corresponding to the eigenvalue 1 is finite-dimensional. As a consequence we get the ?-codimensions for a large amount of arbitrary automorphisms ? of E of order 2.     

Quasi-localizations of ℤ

Motivated by the categorical notion of localizations applied to the quasi-category of abelian groups, we call a homomorphism α: A → B a quasi-localization of abelian groups if for each ϕ ∈ Hom(A,B) there is an n ∈ ℕ and a unique ψ ∈ End(B) such that nϕ = ψ ∘ α. In this case we call B a quasi-localization of A. In this paper we investigate quasi-localizations of the integers ℤ. While it is well-known that localizations of ℤ are just the E-rings, quasi-localizations of ℤ are much more abundant; an injection α: ℤ → M with M torsion-free, is a quasi-localization if and only if, for R = End(M), one has . We call R the ring of the quasi-localization M. Some old results due to Zassenhaus and Butler show that all rings with free additive groups of finite rank are indeed rings of quasi-localizations of ℤ. We will extend this result and show that there are also rings of infinite rank with this property. While there are many realization results of rings R as endomorphism rings of torsion-free abelian groups M in the literature, the group M is usually not contained in the divisible hull of R ⁺, as is required here. We will use a particular case of a category of left R-modules M with a distinguished family of submodules and thus . We will restrict our discussion to the case M = R such that , and in this case we call the family of left ideals E-forcing, not to be confused with the notion of forcing in set theory. We will provide many examples of quasi-localizations M of ℤ, among them those of infinite rank as well as matrix rings for various rings of finite rank.     

When is a 2 × 2 Matrix Ring Over a Commutative Local Ring Quasipolar?

A ring R is quasipolar if for any a ∈ R, there exists p ² = p ∈ R such that p ∈ comm²(a), p + a ∈ U(R) and ap ∈ R ^qnil . In this article, we determine when a 2 × 2 matrix over a commutative local ring is quasipolar. A criterion in terms of solvability of the characteristic equation is obtained for such a matrix to be quasipolar. Consequently, we obtain several equivalent conditions for the 2 × 2 matrix ring over a commutative local ring to be quasipolar. Furthermore, it is shown that the 2 × 2 matrix ring over the ring of p-adic integers is quasipolar.     

Comparison of 3-Dimensional ℤ-Graded and ℤ2-Graded L ∞ Algebras

This article explores L _∞ structures on 3-dimensional vector spaces with both ?- and ?₂-gradings. Since ?-graded L _∞ algebras are special cases of ?₂-graded algebras in the induced ?₂-grading, there are generally fewer ?-graded L _∞ structures on a given space. However, degree zero automorphisms (rather than even automorphisms) determine equivalence in a ?-graded space. We therefore find nontrivial examples in which the map from the ?-graded moduli space to the ?₂-graded moduli space is bijective, injective but not surjective, or surjective but not injective. Additionally, we study how the codifferentials in the moduli spaces deform into other nonequivalent codifferentials.     

Maximal Quotient Rings of Endomorphisms of Quasigenerators

O.Preliminaries. Let R be an associative ring with identity, and let Mod-R denote the category of all unital right R-modules. A set of right ideal of R is called a Gabriel topology on R if satisfies T1. If I∈ and I J, then J∈. T2. If I and J belong to, then I∩J∈. T3. I∈ and r∈R, then (I:r)={x∈R:rx∈I}∈. T4. If I is a right ideal of R and there exists J∈ such that (I:r)∈ for every r∈J, then I∈.     

ℤ/2ℤ Invariants of the free fermion algebra

By using quantum vertex operators we study the invariance of the rank n free-fermion vertex algebra under the action of the group ?∕2? and obtain its minimal generating set. When n = 1, it is well known that this subalgebra is isomorphic to the Virasoro vertex algebra with central charge 1∕2. In the n = 2 case we show that invariant subalgebra is isomorphic to a simple quotient of a certain W-algebra, which we explicitly construct. For n≥3, our approach leads to a rediscovery of the spinor representation of the a?ne vertex algebra associated to the Lie algebra 𝔰𝔬(n) of I. Frenkel.     

On a Regev-Seeman conjecture about ℤ2-graded tensor products

If A,B are superalgebras then, besides A?B, a ?₂-graded tensor product A $ \bar \otimes $ B arises. Kemer proved that if A,B are T-prime algebras then A? B is multi-linear equivalent to a suitable T-prime algebra C. Regev and Seeman conjectured that this holds for A $ \bar \otimes $ B as well. In this paper we prove their conjecture is true indeed, by means of G-graded polynomial identities. The results obtained are valid over any infinite field of characteristic ≠ 2.     

Quotients of the unit ball of ℂn for a free action of ℤ

LetB _n be the unit ball of ℂⁿ and ℤ ≅ Γ ⊂ AutB _n be generated by a parabolic element of AutB _n. We show that the quotientB _n/Γ is biholomorphic to a holomorphically convex domain of ℂⁿ, whose automorphism group is explicity described. It follows thatB _n/ℤ is Stein for any free action of ℤ. Investigation partially supported by University of Bologna. Funds for selected research topics. The second author was supported by an Instituto Nazionale di Alta Matematica grant.     

The Krull–Gabriel Dimension of a Serial Ring

Abstract

We prove that every serial ring R has the isolation property: every isolated point in any theory of modules over R is isolated by a minimal pair. Using this we calculate the Krull–Gabriel dimension of the module category over serial rings. For instance, we show that this dimension cannot be equal to 1.     

The Cardinal of the Set of Non-quasi-regular Elements in a Ring

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The ε c -Product of a Schwartz b-Space by a Quotient Banach Space and Applications

We define the -product of a -space by a quotient Banach space. We give conditions under which this -product will be monic. Finally, we define the _c -product of a Schwartz b-space by a quotient Banach space and we give some examples of applications.     

ℤ2-Graded Number Theory

Let ? be the set of pairs of integers, together with addition and multiplication as given in (1) and (2) below. The arithmetics of ? reflects a certain arithmetics of characters of symmetric groups, whose corresponding Young diagrams are supported on hooks. This arithmetics gives rise to a ?₂-graded (or super or hyperbolic) number theory. Many theorems from number theory have their ?₂-graded analogues in ?. Here we study a few basic aspects of that theory.     

Ordinary and ℤ2-Graded Cocharacters of UT 2(E)

Let E be the infinite dimensional Grassmann algebra over a field F of characteristic 0. In this article we consider the algebra R of 2 × 2 matrices with entries in E and its subalgebra G, which is one of the minimal algebras of polynominal identity (PI) exponent 2. We compute firstly the Hilbert series of G and, as a consequence, we compute its cocharacter sequence. Then we find the Hilbert series of R, using the tool of proper Hilbert series, and we compute its cocharacter sequence. Finally we describe explicitely the ?₂-graded cocharacters of R.     

On Indecomposable Separated Secondary Modules Over ℤ G

In this paper we classify the indecomposable separated secondary modules over G, where G is a cyclic group of prime order and establish a connection between the secondary separated modules and the pure-injective separated modules over such rings.AMS Mathematics Subject Classification (2000) 13F05 16G60