On the injective envelope of a graded module

Abstract

Let Sing _n be the semigroup of all singular full transformations on the set X _n  = {1, 2,…, n} under the composition of functions. Let E(J _n ? 1) be the set of all idempotents of the top 𝒥-class J _n ? 1 = {α ∈ Sing _n :|im α| = n-1}. For any nonempty subset I of E(J _n  ? 1), the aim of this paper is to find a constructive necessary and sufficient condition for the semiband S(I) = ?I? to be ?-trivial. Further, the semiband S(I) is locally maximal ?-trivial if S(I) is ?-trivial and S(I ∪ {e}) is not ?-trivial for any e ∈ E(J _n ? 1 )\I. As applications, we classify locally maximal ?-trivial subsemibands and locally maximal regular ?-trivial subsemibands of Sing _n , respectively. Moreover, the characterization of which S(I) is a band is obtained.

     

On the Simplicity of Lie Algebras Associated to Leavitt Algebras

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L _𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M _d (L _𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ^?, and consider its Lie subalgebra [S ^?, S ^?]. In our main result, we show that [S ^?, S ^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L _𝕂(n)^?, L _𝕂(n)^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.     

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.     

Decompositions of Quotient Rings and m-Power Commuting Maps

Let R be a semiprime ring with symmetric Martindale quotient ring Q, n ≥ 2 and let f(X) = X ⁿ h(X), where h(X) is a polynomial over the ring of integers with h(0) = ±1. Then there is a ring decomposition Q = Q ₁ ⊕ Q ₂ ⊕ Q ₃ such that Q ₁ is a ring satisfying S _2n?2, the standard identity of degree 2n ? 2, Q ₂ ? M _n (E) for some commutative regular self-injective ring E such that, for some fixed q > 1, x ^q  = x for all x ∈ E, and Q ₃ is a both faithful S _2n?2-free and faithful f-free ring. Applying the theorem, we characterize m-power commuting maps, which are defined by linear generalized differential polynomials, on a semiprime ring.     

A Generalization of a Lemma of Sullivan

Consider an irreducible polynomial of the form f(X) = X ^p  ? aX ? b ∈ 𝔽[X] and α a root of f(X), where 𝔽 is a field of characteristic p. In 1975, F.J. Sullivan stated a lemma that provides the trace, taken with respect to the extension 𝔽(α)/𝔽, of elements of the form α ⁿ , where 0 ≤ n ≤ p ² ? 1. We present a generalization of Sullivan's Lemma and provide another proof of the original lemma. We explain how computing Tr(α ⁿ ) for n < p ^r can be reduced to computing the traces Tr(α ^m ) for all m ≤ r(p ? 1).     

On 𝔬𝔰𝔭(1 | 2)-Relative Cohomology on S 1|1

We compute the second 𝔬𝔰𝔭(1 | 2) ?relative cohomology space of 𝒦(1) with coefficients in the module of λ-densities 𝔉_λ on S ^1|1. This result allows us to compute the second 𝔬𝔰𝔭(1 | 2) ?relative cohomology space of 𝒦(1) with coefficients in the Poisson superalgebra 𝒮𝒫. We explicitly give 2-cocycles spanning these cohomology spaces.     

Range-Inclusive Maps in Rings with Idempotents

Let 𝒜 be a ring, let ? be an 𝒜-bimodule, and let 𝒞 be the center of ?. A map F:𝒜 → ? is said to be range-inclusive if [F(x), 𝒜] ? [x, ?] for every x ∈ 𝒜. We show that if 𝒜 contains idempotents satisfying certain technical conditions (which we call wide idempotents), then every range-inclusive additive map F:𝒜 → ? is of the form F(x) = λx + μ(x) for some λ ∈ 𝒞 and μ:𝒜 → 𝒞. As a corollary we show that if 𝒜 is a prime ring containing an idempotent different from 0 and 1, then every range-inclusive additive map from 𝒜 into itself is commuting (i.e., [F(x), x] = 0 for every x ∈ 𝒜).     

The multiplier ideal is a universal test ideal

Abstract

For a canonical threefold X, we know that h ⁰(X, 𝒪 _X (nK _X )) ≥ 1 for a sufficiently large n. When χ(𝒪 _X ) > 0, it is not easy to get such an integer n. Fletcher showed that h ⁰(X, 𝒪 _X (12K _X )) ≥ 1 and h ⁰(X, 𝒪 _X (24K _X )) ≥ 2 when χ(𝒪 _X ) = 1. He inquired about existence of a canonical threefold with given conditions which shows the result sharp. We show that such an example does not exist. Using a different technique, we prove h ⁰(X, 𝒪 _X (12K _X )) ≥ 2.     

Orthogonal Groups 𝒪(n) over GF(2) as Automorphisms

Let (V, Q) be a quadratic vector space over a fixed field. Orthogonal group 𝒪(V, Q) is defined as automorphisms on (V, Q). If Q = I, it is 𝒪(V, I) = 𝒪(n). There is a nice result that 𝒪(n) ? Aut(𝔬(n)) over ? or ?, where 𝔬(n) is the Lie algebra of n × n alternating matrices over the field. How about another field The answer is “Yes” if it is GF(2). We show it explicitly with the combinatorial basis ?. This is a verification of Steinberg's main result in 1961, that is, Aut(𝔬(n)) is simple over the square field, with a nonsimple exception Aut(𝔬(5)) ? 𝒪(5) ? 𝔖₆.     

Multiplicative Lie Isomorphisms Between Prime Rings

Let ? be a prime ring with 1 containing a nontrivial idempotent E, and let ?′ be another prime ring. If Φ:? → ?′ is a multiplicative Lie isomorphism, then Φ(T + S) = Φ(T) + Φ(S) + Z′ _T,S for all T, S ∈ ?, where Z′ _T,S is an element in the center 𝒵′ of ?′ depending on T and S.     

Some Results for Decomposition Numbers for the Hecke Algebra of Type A with q = −1

Let ? = ?_{?, ?1}(𝔖 _n ) be the Hecke algebra of the symmetric group 𝔖 _n . For partitions λ and ν with ν 2 ? regular, define the Specht module S(λ) and the irreducible module D(ν). Define d _λν = [S(λ): D(ν)] to be the composition multiplicity of D(ν) in S(λ). In this paper we compute the decomposition numbers d _λν for all partitions of the form λ = (a, c, 1 ^b ) and ν 2 ? regular.     

Adjacent Ideals to Simple Complete Ideals in Two-Dimensional Regular Local Rings

Let R be a two-dimensional regular local ring with infinite residue field, and ? be a simple complete residually rational ideal of R of order r which determines R _h . Let 𝒯 be the set of quadratic transforms T of R _h with [T: R _h ] = 1, and 𝒮 the set of simple complete ideals of R of order r which are adjacent to ? from below. If R _h is free respectively a satellite, then there exist T* ∈ 𝒯 respectively T*, T** ∈ 𝒯 and a bijective map between the set 𝒮 and the set 𝒯?{T*} respectively 𝒯?{T*, T**}.     

PRESENTATIONS FOR SOME MONOIDS OF PARTIAL TRANSFORMATIONS ON A FINITE CHAIN

ABSTRACT

In this paper we calculate presentations for some natural monoids of transformations on a chain X _n  = {1 < 2 <?s < n}. First we consider 𝒪𝒟 _n [𝒫𝒪𝒟 _n ], the monoid of all full [partial] transformations on X _n that preserve or reverse the order. Two other monoids of partial transformations on X _n we look at are 𝒫𝒪𝒫 _n and 𝒫𝒪? _n –-the elements of the first preserve the orientation and the elements of the second preserve or reverse the orientation.     

Group Laws [x,y −1] ≡ u(x,y) and Varietal Properties

Let F = ?x, y? be a free group. It is known that the commutator [x, y ^?1] cannot be expressed in terms of basic commutators, in particular in terms of Engel commutators. We show that the laws imposing such an expression define specific varietal properties. For a property 𝒫 we consider a subset U(𝒫) ? F such that every law of the form [x, y ^?1] ≡ u, u ∈ U(𝒫) provides the varietal property 𝒫. For example, we show that each subnormal subgroup is normal in every group of a variety 𝔙 if and only if 𝔙 satisfies a law of the form [x, y ^?1] ≡ u, where u ∈ [F′, ?x?].     

A GENERALIZATION OF AUSLANDER–BUCHSBAUM AND BASS FORMULAS

Abstract

Given a contravariant functor F : 𝒞 → 𝒮ets for some category 𝒞, we say that F (𝒞) (or F) is generated by a pair (X, x) where X is an object of 𝒞 and x ∈ F(X) if for any object Y of 𝒞 and any y ∈ F(Y), there is a morphism f : Y → X such that F(f)(x) = y. Furthermore, when Y = X and y = x, any f : X → X such that F(f)(x) = x is an automorphism of X, we say that F is minimally generated by (X, x). This paper shows that if the ring R is left noetherian, then there exists a minimal generator for the functor ?xt (?, M) : ? → 𝒮ets, where M is a left R-module and ? is the class (considered as full subcategory of left R-modules) of injective left R-modules.     

Toric Rings Arising from Cyclic Polytopes

Let d and n be positive integers with n ≥ d + 1 and 𝒫 ? ? ^d an integral cyclic polytope of dimension d with n vertices, and let K[𝒫] = K[?_≥0𝒜_𝒫] denote its associated semigroup K-algebra, where 𝒜_𝒫 = {(1, α) ∈ ? ^d+1: α ∈ 𝒫} ∩ ? ^d+1 and K is a field. In the present paper, we consider the problem when K[𝒫] is Cohen–Macaulay by discussing Serre's condition (R ₁), and we give a complete characterization when K[𝒫] is Gorenstein. Moreover, we study the normality of the other semigroup K-algebra K[Q] arising from an integral cyclic polytope, where Q is a semigroup generated by its vertices only.     

Length Four Polynomial Automorphisms

We study the structure of length four polynomial automorphisms of R[X, Y] when R is a unique factorization domain. The results from this study are used to prove that, if SL _m (R[X ₁, X ₂,…, X _n ]) = E _m (R[X ₁, X ₂,…, X _n ]) for all n, m ≥ 0, then all length four polynomial automorphisms of R[X, Y] that are commutators are stably tame.