The prüfer rings that are endomorphism rings of artinian modules

In this paper we characterize the (commutative) Priifer rings that can be realized as endomorphism rings of artinian modules over arbitrary associative rings with identity (Theorem 4.7). This characterization is obtained by determining the structure of ∑-pure-injective modules over Prufer rings (Theorems 3.4 and 3.5)     

Twisted rings and moduli stacks of “fat” point modules in non-commutative projective geometry

The Hilbert scheme of point modules was introduced by Artin–Tate–Van den Bergh to study non-commutative graded algebras. The key tool is the construction of a map from the algebra to a twisted ring on this Hilbert scheme. In this paper, we study moduli stacks of more general “fat” point modules, and show that there is a similar map to a twisted ring associated to the stack. This is used to provide a sufficient criterion for a non-commutative projective surface to be birationally PI. It is hoped that such a criterion will be useful in understanding Mike Artin?s conjecture on the birational classification of non-commutative surfaces.     

Every order is the endomorphism ring of a projective module over a quasi–hereditary order

We give in this note some new bounds on p _c(G) in terms of σ _c for several classes of finite groups, in particular, we prove that ρ _c ≤4σ _c (G) for any finite solvable group G, which improves some known results. We also pose some related open problems.     

Gr?bner bases in difference-differential modules with coefficients in a commutative ring

Insa and Pauer presented a basic theory of Grbner bases for differential operators with coefficients in a commutative ring and an improved version of this result was given by Ma et al.In this paper,we present an algorithmic approach for computing Grbner bases in difference-differential modules with coefficients in a commutative ring.We combine the generalized term order method of Zhou and Winkler with SPoly method of Insa and Pauer to deal with the problem.Our result is a generalization of theories of Insa and Pauer,Ma et al.,Zhou and Winkler and includes them as special cases.     

A convexity property and a new characterization of Euler’s gamma function

Our main results are:

1. Let α ≠ 0 be a real number. The function (Γ ? exp) ^α is convex on ${\mathbf{R}}$ if and only if $$\alpha \geq \max_{0<{t}<{x_0}}\Big(-\frac{1}{t\psi(t)} - \frac{\psi'(t)}{\psi(t)^2}\Big) = 0.0258... .$$ Here, x ₀ = 1.4616... denotes the only positive zero of ${\psi = \Gamma'/\Gamma}$ .

1. Assume that a function f: (0, ∞) → (0, ∞) is bounded from above on a set of positive Lebesgue measure (or on a set of the second category with the Baire property) and satisfies $$f(x+1) = x f(x) \quad{\rm for}\quad{x > 0}\quad{\rm and}\quad{f(1) = 1}.$$

If there are a number b and a sequence of positive real numbers (a _n ) ${(n \in \mathbf{N})}$ with ${{\rm lim}_{n\to\infty} a_n =0}$ such that for every n the function ${(f \circ {\rm exp})^{a_n}}$ is Jensen convex on (b, ∞), then f is the gamma function.     

A “quite superfluous” characterization of the Jacobson radical of an associative ring

It is well-known that the Jacobson radical of a unital ring R is the largest superfluous right ideal of R. While a simple example shows this to be false for non-unital rings, it is shown that the result remains valid in the absence of a unit provided the notion of “superfluous” is taken relative to all regular right ideals. The purpose of this note is to provide a proof for this little-known fact.     

A new proof for classification of irreducible modules of a Hecke algebra of type An−1

In this paper we give a new proof for the classification of irreducible modules of a Hecke algebra of type $A_{n?1}$, which was obtained by Dipper and James in 1986.     

Normalizer of a group of “diagonal” automorphisms in algebras over a commutative ring

Let S be a faithful algebra over commutative ring R. It is assumed that S is additively generated by its invertible elements. It is shown that the nomalizer of subgroup Aut(S_s) of group Aut(S_R) coincides with the semidirect product Aut(S_S) Aut(S/R),where the second factor is the group of all ring automorphisms of ring S identical on R.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 191, pp. 5–8, 1991.     

Moore–Penrose invertibility of differences and products of projections in rings with involution

This article concerns the MP inverse of the differences and the products of projections in a ring R   with involution. Some equivalent conditions are obtained. As applications, the MP invertibility of the commutator pq−qp$pq-qp$ and the anti-commutator pq+qp$pq+qp$ are characterized, where p and q are projections in R  . Some related known results in C^?$C^{?}$-algebra are generalized.     

A counterexample to the paper ��Weakly associated primes and primary decomposition of modules over commutative rings��

In this note we will show that finitely generated condition is necessary in Theorem 1.2 of the above mentioned paper.     

Classification of Irreducible Weight Modules with a Finite-dimensional Weight Space over Twisted Heisenberg-Virasoro Algebra

We show that the support of an irreducible weight module over the twisted Heisenberg-Virasoro algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the twisted Heisenber-Virasoro algebra, having a nontrivial finite-dimensional weight space, is a Harish-Chandra module （and hence is either an irreducible highest or lowest weight module or an irreducible module from the intermediate series）.     

The Category of Maximal Cohen–Macaulay Modules as a Ring with Several Objects

Over a commutative local Cohen–Macaulay ring, we view and study the category of maximal Cohen–Macaulay modules as a ring with several objects. We compute the global dimension of this category and thereby extend some results of Iyama and Leuschke.     

A new class of unique product monoids with applications to ring theory

We introduce a class of ordered monoids defined by the existence of certain “unique products” with respect to artinian and narrow subsets of the monoid. The logical relationships between this and other significant classes of monoids are explicated with several examples. We conclude with results on skew generalized power series rings. The new class of monoids provides the appropriate setting for obtaining results on reduced rings and domains of skew generalized power series, and on analogues of Armendariz rings.     

Metric Möbius geometry and a characterization of spheres

We obtain a Möbius characterization of the n-dimensional spheres S ⁿ endowed with the chordal metric d ₀. We show that every compact extended Ptolemy metric space with the property that every three points are contained in a circle is Möbius equivalent to (S ⁿ , d ₀) for some n ≥ 1.     

Cyclic modules with ∞-simplicial faces and the cyclic homology of A ∞-algebras

Mathematical Notes - A chain bicomplex for A ∞-algebras, which generalizes the Tsygan chain bicomplex in the theory of cyclic homology of associative algebras, is constructed by using the...     

The flow of a supersonic ideal gas with “weak” and “strong” shocks over a wedge

The problem of the flow of a uniform supersonic ideal (inviscid and non-heat-conducting) gas over a wedge is considered. If the turning angle of the flow, which is equal to the angle of inclination of the generatrix of the wedge, is less than the maximum value, the problem has two solutions. In the solution with an oblique low-intensity (“weak”) shock, the uniform flow between the shock and the wedge is almost always supersonic. One exception is a small vicinity of the maximum turning angle. For an ideal gas this vicinity does not exceed a fraction of a degree at all Mach numbers. Behind a high-intensity (“strong”) shock, the flow of an ideal gas is always subsonic. “Weak” shocks are observed in all experiments with finite wedges. Some researchers attribute this preference to the “downstream” boundary conditions (“on the right at infinity” for a flow incident on the wedge from the left), and others attribute it to the instability (“Lyapunov” instability) of a flow with a strong shock when it flows over the wedge and to the stability of flow with a weak shock. The results presented below from calculations of the flows that occur for finite wedges within the two-dimensional unsteady Euler equations, when the parameters behind the strong shock are specified on the right-hand boundary, i.e., on the arc of a circle between the wedge and the shock, demonstrate the correctness of the conclusion of the first group of researchers and the incorrectness of the conclusion of the other group. In these calculations, after both small and fairly large perturbations, the flows investigated (which are, in fact, Lyapunov unstable!) return to the solution with a strong shock. In addition, the problem of steady flow over a wedge was regarded as the limit of the two-dimensional non-steady problems at infinite time. Simplification of one of them leads to the problem of the submerged over-expanded supersonic steady outflow. In the ideal gas model this problem is equivalent to flow over a wedge with both weak and strong shocks. All the solutions considered are stable.     

The Piltz’s problem over a ring of matrices of second order with integer elements

The asymptotic formula for a summatory function of the number of representations of matrices from $ {M_2}\left( \mathbb{Z} \right) $ in the form C = A ₁ A ₂ A ₃ under the condition det C ≤ x is constructed. The second moment of the error term of this asymptotic formula is estimated.