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.     

Irreducibility Criterion Over Finite Fields

The aim of this article is to prove the irreducibility of the polynomial Λ(Y) = Y ^d  + λ _d?1 Y ^d?1 + … + λ₀ over 𝔽 _q [X] where λ _i ∈ 𝔽 _q [X] and deg λ _d?1 > deg λ _i for each i ≠ d ? 1. We discuss in particular connections between the irreducible polynomials Λ and the number of Pisot elements in the case of formal power series.     

Annihilators of Skew Derivations with Engel Conditions on Lie Ideals

Let R be a prime ring, L a noncentral Lie ideal of R, and a ∈ R. Set [x, y]₁ = [x, y] = xy ? yx for x, y ∈ R and inductively [x, y]_k = [[x, y]_k?1, y] for k > 1. Suppose that δ is a nonzero σ-derivation of R such that a[δ(x), x]_k = 0 for all x ∈ L, where σ is an automorphism of R and k is a fixed positive integer. Then a = 0 except when char R = 2 and R ? M₂(F), the 2 × 2 matrix ring over a field F.     

Depth of Initial Ideals of Normal Edge Rings

Let G be a finite graph on the vertex set [d] = {1,…, d} with the edges e ₁,…, e _n and K[t] = K[t ₁,…, t _d ] the polynomial ring in d variables over a field K. The edge ring of G is the semigroup ring K[G] which is generated by those monomials t ^e  = t _i t _j such that e = {i, j} is an edge of G. Let K[x] = K[x ₁,…, x _n ] be the polynomial ring in n variables over K, and define the surjective homomorphism π: K[x] → K[G] by setting π(x _i ) = t ^{e _i} for i = 1,…, n. The toric ideal I _G of G is the kernel of π. It will be proved that, given integers f and d with 6 ≤ f ≤ d, there exists a finite connected nonbipartite graph G on [d] together with a reverse lexicographic order <_rev on K[x] and a lexicographic order <_lex on K[x] such that (i) K[G] is normal with Krull-dim K[G] = d, (ii) depth K[x]/in_<rev (I _G ) = f and K[x]/in_<lex (I _G ) is Cohen–Macaulay, where in_<rev (I _G ) (resp., in_<lex (I _G )) is the initial ideal of I _G with respect to <_rev (resp., <_lex) and where depth K[x]/in_<rev (I _G ) is the depth of K[x]/in_<rev (I _G ).     

Generalized Derivations with Engel Conditions on Polynomials

Let R be a noncommutative prime ring with extended centroid C, and let D: R → R be a nonzero generalized derivation, f(X ₁,…, X _t ) a nonzero polynomial in noncommutative indeterminates X ₁,…, X _t over C with zero constant term, and k ≥ 1 a fixed integer. In this article, D and f(X ₁,…, X _t ) are characterized if the Engel identity is satisfied: [D(f(x ₁,…, x _t )), f(x ₁,…, x _t )] _k  = 0 for all x ₁,…, x _t  ∈ R.     

Finitely Presented Functors and Degenerations#

Given two d-dimensional Λ-modules M and N, then M degenerates to N if and only if there exists an exact sequence of the form 0→ U→ U ⊕ M→ N→ 0 for some U ? mod Λ (Zwara, 1998 Zwara , G. ( 1998 ). A degeneration-like order for modules . Arch. Math. 71 : 437 – 444 . [CSA] [CROSSREF]  [Google Scholar]). Having this as a starting point, in this article we give a characterization of degenerations by the existence of a certain finitely presented functor. This gives new information about U in the sequence above. We show how this new information can be used to prove that M   _deg N even when M ⊕ X ≤  _deg N ⊕ X for some X for modules over Λ _q  = k〈 x,y〉/〈 x ²,y ²,xy + qyx〉, q ≠ 0 in k, where k is an algebraically closed field.     

Powers of Elements and Monomial Ideals #

All monomial ideals I ? k[X ₀,…, X _d ] are classified which satisfy the following condition: If f ∈ I with f ⁿ  ∈ I ⁿ⁺¹ for some n, then f ∈ (X ₀,…, X _d ) I.     

Crossed products and blocks with normal defect groups

Abstract

Let D be an integral domain. A multiplicative set S of D is an almost splitting set if for each 0 ≠ d ∈ D, there exists an n = n(d) with d ⁿ  = st where s ∈ S and t is v-coprime to each element of S. An integral domain D is an almost GCD (AGCD) domain if for every x, y ∈ D, there exists a positive integer n = n(x, y) such that x ⁿ D ∩ y ⁿ D is a principal ideal. We prove that the polynomial ring D[X] is an AGCD domain if and only if D is an AGCD domain and D[X] ? D′[X] is a root extension, where D′ is the integral closure of D. We also show that D + XD _S [X] is an AGCD domain if and only if D and D _S [X] are AGCD domains and S is an almost splitting set.     

Prophet Regions for Independent [0, 1]-Valued Random Variables with Random Discounting

Abstract

Let X ₁, X ₂… and B ₁, B ₂… be mutually independent [0, 1]-valued random variables, with EB _j  = β > 0 for all j. Let Y _j  = B ₁ … sB _j?1 X _j for j ≥ 1. A complete comparison is made between the optimal stopping value V(Y ₁,…,Y _n ):=sup{EY _τ:τ is a stopping rule for Y ₁,…,Y _n } and E(max _1≤j≤n Y _j ). It is shown that the set of ordered pairs {(x, y):x = V(Y ₁,…,Y _n ), y = E(max _1≤j≤n Y _j ) for some sequence Y ₁,…,Y _n obtained as described} is precisely the set {(x, y):0 ≤ x ≤ 1, x ≤ y ≤ Ψ _{n, β}(x)}, where Ψ _{n, β}(x) = [(1 ? β)n + 2β]x ? β^?(n?2) x ² if x ≤ β ^n?1, and Ψ _{n, β}(x) = min _j≥1{(1 ? β)jx + β ^j } otherwise. Sharp difference and ratio prophet inequalities are derived from this result, and an analogous comparison for infinite sequences is obtained.     

X-Inner Automorphisms,Automorphisms, and Isomorphisms of Skew Polynomial Rings

Abstract

In this paper, we examine the X-inner automorphisms, automorphisms, and isomorphisms of skew polynomial rings of the form K[x][y;δ], where K is a field of characteristic 0 and δ is a derivation of K [x] such that x ^δ is a polynomial of degree ≥ 1. We 1.  determine the group of X-inner automorphisms of K [x][ y;δ],

2.  analyze the structure of the group of automorphisms of K [x][ y;δ], and

3.  examine the isomorphism classes of skew polynomial rings of the form K [x][ y;δ].

We also provide several examples which indicate the importance of the base field in computing the X-inner automorphisms, automorphisms, and isomorphism classes of K [x][ y;δ].     

Catenarian Property of Mixed Polynomial and Power Series Rings Over a Pullback

Let T be an integral domain with a maximal ideal M, ?: T → K: = T/M the natural surjection, and R the pullback ?^?1(D), where D is a proper subring of K. We give necessary and sufficient conditions for the mixed extensions R[x ₁]]…[x _n ]] to be catenarian, where each [x _i ]] is fixed as either [x _i ] or [[x _i ]]. We also give a complete answer to the question of determining the field extensions k ? K such that the contraction map Spec(K[x ₁]]…[x _n ]]) → Spec(k[x ₁]]…[x _n ]]) is a homeomorphism. As an application, we characterize the globalized pseudo-valuation domains R such that R[x ₁]]…[x _n ]] is catenarian.     

Prime T-Ideals in Polynomial and Power Series Rings Over a Pseudo-Valuation Domain

Let R be an m-dimensional pseudo-valuation domain with residue field k, let V be the associated valuation domain with residue field K, and let k ₀ be the maximal separable extension of k in K. We compute the t-dimension of polynomial and power series rings over R. It is easy to see that t-dim R[x ₁,…, x _n ] = 2 if m = 1 and K is transcendental over k, but equals m otherwise, and that t-dim R[[x ₁,…, x _n ]] = ∞ if R is a nonSFT-ring. When R is an SFT-ring, we also show that: (1) t-dim R[[x]] = m; (2) t-dim R[[x ₁,…, x _n ]] = 2m ? 1, if n ≥ 2, K has finite exponent over k ₀, and [k ₀: k] < ∞; (3) t-dim R[[x ₁,…, x _n ]] = 2m, otherwise.     

Cohomology of general coinduced representions

ABSTRACT

Let R be a prime ring with a nonzero derivation d and let f(X ₁,…,X _t ) be a multilinear polynomial over C, the extended centroid of R. Suppose that b[d(f(x ₁,…,x _t )), f(x ₁,…,x _t )] _n  = 0 for all x _i  ∈ R, where 0 ≠ b ∈ R and n is a fixed positive integer. Then f(X ₁,…,X _t ) is centrally valued on R unless char R = 2 and dim _C RC = 4. We prove a more generalized version by replacing R with a left ideal.     

The Degree and Regularity of Vanishing Ideals of Algebraic Toric Sets Over Finite Fields

Let X* be a subset of an affine space 𝔸 ^s , over a finite field K, which is parameterized by the edges of a clutter. Let X and Y be the images of X* under the maps x → [x] and x → [(x, 1)], respectively, where [x] and [(x, 1)] are points in the projective spaces ? ^s?1 and ? ^s , respectively. For certain clutters and for connected graphs, we were able to relate the algebraic invariants and properties of the vanishing ideals I(X) and I(Y). In a number of interesting cases, we compute its degree and regularity. For Hamiltonian bipartite graphs, we show the Eisenbud–Goto regularity conjecture. We give optimal bounds for the regularity when the graph is bipartite. It is shown that X* is an affine torus if and only if I(Y) is a complete intersection. We present some applications to coding theory and show some bounds for the minimum distance of parameterized linear codes for connected bipartite graphs.     

Closed Polynomials in Polynomial Rings over Unique Factorization Domains

Let R be a UFD, and let M(R, n) be the set of all subalgebras of the form R[f], where f ∈ R[x ₁,…, x _n ]?R. For a polynomial f ∈ R[x ₁,…, x _n ]?R, we prove that R[f] is a maximal element of M(R, n) if and only if it is integrally closed in R[x ₁,…, x _n ] and Q(R)[f] ∩ R[x ₁,…, x _n ] = R[f]. Moreover, we prove that, in the case where the characteristic of R equals zero, R[f] is a maximal element of M(R, n) if and only if there exists an R-derivation on R[x ₁,…, x _n ] whose kernel equals R[f].     

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.