Prime Ideals in Algebras Determined by Submonoids of Nilpotent Groups

The prime spectrum of the semigroup algebra K[S] of a submonoid S of a finitely generated nilpotent group is studied via the spectra of the monoid S and of the group algebra K[G] of the group G of fractions of S. It is shown that the classical Krull dimension of K[S] is equal to the Hirsch length of the group G provided that G is nilpotent of class two. This uses the fact that prime ideals of S are completely prime. An infinite family of prime ideals of a submonoid of a free nilpotent group of class three with two generators which are not completely prime is constructed. They lead to prime ideals of the corresponding algebra. Prime ideals of the monoid of all upper triangular n × n matrices with non-negative integer entries are described and it follows that they are completely prime and finite in number.     

The endomorphism spectrum of a monounary algebra

The endomorphism spectrum specA of an algebra A is defined as the set of all positive integers, which are equal to the number of elements in an endomorphic image of A, for all endomorphisms of A. In this paper we study finite monounary algebras and their endomorphism spectrum. If a finite set S of positive integers is given, one can look for a monounary algebra A with S = specA. We show that for countably many finite sets S, no such A exists. For some sets S, an appropriate A with spec A = S are described. For n ∈ ? it is easy to find a monounary algebra A with {1, 2, ..., n} = specA. It will be proved that if i ∈ ?, then there exists a monounary algebra A such that specA skips i consecutive (consecutive eleven, consecutive odd, respectively) numbers. Finally, for some types of finite monounary algebras (binary and at least binary trees) A, their spectrum is shown to be complete.     

The set of irreducible graphs for the projective plane is finite

In 1930 Kuratowski proved that a graph does not embed in the real plane R² if and only if it contains a subgraph homeomorphic to one of two graphs, K₅ or K_{3, 3}. For positive integer n, let I_n (P) denote a smallest set of graphs whose maximal valency is n and such that any graph which does not embed in the real projective plane contains a subgraph homeomorphic to a graph in I_n (P) for some n. Glover and Huneke and Milgram proved that there are only 6 graphs in I₃ (P), and Glover and Huneke proved that I_n (P) is finite for all n. This note proves that I_n (P) is empty for all but a finite number of n. Hence there is a finite set of graphs for the projective plane analogous to Kuratowski's two graphs for the plane.     

André-Quillen homology of algebra retracts

Given a homomorphism of commutative noetherian rings ?:R→S, Daniel Quillen conjectured in 1970 that if the André-Quillen homology functors D_n(S∣R;−) vanish for all n?0, then they vanish for all n?3. We prove the conjecture under the additional hypothesis that there exists a homomorphism of rings ψ:S→R such that ?°ψ=id_S. More precisely, in this case we show that ψ is a complete intersection at for every prime ideal  of S. Using these results, we describe all algebra retracts S→R→S for which the algebra Tor_•^R(S,S) is finitely generated over Tor₀^R(S,S)=S.     

Ideals in tensor powers of the enveloping algebra u(sl 2)

Let U = U(_sl₂)^?n be the tensor power of n copies of the enveloping algebra U(sl ₂) over an arbitrary field K of characteristic zero. In this paper we list the prime ideals of U by generators and classify them by height. If Z is the center of U and J is a prime ideal of Z, there are exactly 2₅ prime ideals I of U with I ∩ Z = J, where 0 ≤ s = s(J) ≤ n is an integer. Indeed, with respect to inclusion, they form a lattice isornorphic to the lattice of subsets of a set. When J is a maximal ideal of Z, there are only finitely many two-sided ideals of U containing J, They are presented by generators and their lattice is described, In particular, for each such J there exists a unique maximal ideal of U containing J and a unique ideal of U minimal with respect to the property that it properly contains JU. Similar results are given in the case when U is the tensor product of infinitely many copies of U(sl ₂).     

Cyclic irreducible non-holonomic modules over the Weyl algebra: An algorithmic characterization

Let K be a field of characteristic zero, n≥1 an integer and A_n+1=K[X,Y₁,…,Y_n]〈∂_X,∂_Y1,…,∂_Yn〉 the (n+1)th Weyl algebra over K. Let S∈A_n+1 be an order-1 differential operator of the type with a_i,b_i∈K[X] and g_i∈K[X,Y_i] for every i=1,…,n. We construct an algorithm that allows one to recognize whether S generates a maximal left ideal of A_n+1, hence also whether A_n+1/A_n+1S is an irreducible non-holonomic A_n+1-module. The algorithm, which is a powerful instrument for producing concrete examples of cyclic maximal left ideals of A_n, is easy to implement and quite useful; we use it to solve several open questions.The algorithm also allows one to recognize whether certain families of algebraic differential equations have a solution in K[X,Y₁,…,Y_n] and, when they have one, to compute it.     

Torsion theories induced from commutative subalgebras

We begin a study of torsion theories for representations of finitely generated algebras U over a field containing a finitely generated commutative Harish-Chandra subalgebra Γ. This is an important class of associative algebras, which includes all finite W-algebras of type A over an algebraically closed field of characteristic zero, in particular, the universal enveloping algebra of (or ) for all n. We show that any Γ-torsion theory defined by the coheight of the prime ideals of Γ is liftable to U. Moreover, for any simple U-module M, all associated prime ideals of M in have the same coheight. Hence, the coheight of these associated prime ideals is an invariant of a given simple U-module. This implies the stratification of the category of U-modules controlled by the coheight of the associated prime ideals of Γ. Our approach can be viewed as a generalization of the classical paper by Block (1981) [4]; it allows, in particular, to study representations of beyond the classical category of weight or generalized weight modules.     

Reducibility number

Let P be a poset in a class of posets P. A smallest positive integer r is called reducibility number of P with respect to P if there exists a non-empty subset S of P with |S|=r and P-S∈P. The reducibility numbers for the power set 2ⁿ of an n-set (n?2) with respect to the classes of distributive lattices, modular lattices and Boolean lattices are calculated. Also, it is shown that the reducibility number r of the lattice of all subgroups of a finite group G with respect to the class of all distributive lattices is 1 if and only if the order of G has at most two distinct prime divisors; further if r is a prime number then order of G is divisible by exactly three distinct primes. The class of pseudo-complemented u-posets is shown to be reducible. Deletable elements in semidistributive posets are characterized.     

Note on bounds for multiplicities

Let S=K[x₁,…,x_n] be a polynomial ring and R=S/I be a graded K-algebra where I⊂S is a graded ideal. Herzog, Huneke and Srinivasan have conjectured that the multiplicity of R is bounded above by a function of the maximal shifts in the minimal graded free resolution of R over S. We prove the conjecture in the case that codim(R)=2 which generalizes results in (J. Pure Appl. Algebra 182 (2003) 201; Trans. Amer. Math. Soc. 350 (1998) 2879). We also give a proof for the bound in the case in which I is componentwise linear. For example, stable and squarefree stable ideals belong to this class of ideals.     

Set theoretic complete intersection for curves in a smooth affine algebra

Let A be a regular ring of dimension d (d≥3) containing an infinite field k. Let n be an integer such that 2n≥d+3. Let I be an ideal in A of height n and P be a projective A-module of rank n. Suppose P⊕A≈Aⁿ⁺¹ and there is a surjection α: P→I. It is proved in this note that I is a set theoretic complete intersection ideal. As a consequence, a smooth curve in a smooth affine C-algebra with trivial conormal bundle is a set theoretic complete intersection if its corresponding class in the Grothendieck group is torsion.     

The commuting graphs of some subsets in the quaternion algebra over the ring of integers modulo <Emphasis Type="Italic">N</Emphasis>

Let R be an arbitrary ring, S be a subset of R, and Z(S) = {s ∈ S | sx = xs for every x ∈ S}. The commuting graph of S, denoted by Γ(S), is the graph with vertex set S \ Z(S) such that two different vertices x and y are adjacent if and only if xy = yx. In this paper, let I _n , N _n be the sets of all idempotents, nilpotent elements in the quaternion algebra ℤ _n [i, j, k], respectively. We completely determine Γ(I _n ) and Γ(N _n ). Moreover, it is proved that for n ≥ 2, Γ(I _n ) is connected if and only if n has at least two odd prime factors, while Γ(N _n ) is connected if and only if n ∈ 2, 2², p, 2p for all odd primes p.     

A procedure to compute prime filtration

Let K be a field, S = K[x ₁, … x _n ] be a polynomial ring in n variables over K and I ⊂ S be an ideal. We give a procedure to compute a prime filtration of S/I. We proceed as in the classical case by constructing an ascending chain of ideals of S starting from I and ending at S. The procedure of this paper is developed and has been implemented in the computer algebra system Singular.     

Temperley-Lieb planar algebra modules arising from the ADE planar algebras

A Hilbert module over a planar algebra P is essentially a Hilbert module over a canonically defined algebra spanned by the annular tangles in P. It follows that any planar algebra Q containing P is a module over P, and in particular, any subfactor planar algebra is a module over the Temperley-Lieb planar algebra with the same modulus. We describe a positivity result that allows us to describe irreducible Temperley-Lieb planar algebra modules, and apply the result to decompose the planar algebras determined by the Coxeter graphs A_n (n?3), D_n (n?4), E₆, E₇, and E₈.     

On extensions of ideals in posets

Two kinds of order ideals, namely, the n-prime ideals and k-closed ideals of a poset are introduced and shown to be equivalent in the sense that an order-ideal is (n+1)-prime if and only if it is n-closed. Also a prime radical theorem for these ideals is proved.     

Sklyanin algebras and Hilbert schemes of points

We construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P².The generic noncommutative plane corresponds to the Sklyanin algebra S=Skl(E,σ) constructed from an automorphism σ of infinite order on an elliptic curve E⊂P². In this case, the fine moduli space of line bundles over S with first Chern class zero and Euler characteristic 1−n provides a symplectic variety that is a deformation of the Hilbert scheme of n points on P²?E.     

A Prime Ideal Principle in commutative algebra

In this paper, we offer a general Prime Ideal Principle for proving that certain ideals in a commutative ring are prime. This leads to a direct and uniform treatment of a number of standard results on prime ideals in commutative algebra, due to Krull, Cohen, Kaplansky, Herstein, Isaacs, McAdam, D.D. Anderson, and others. More significantly, the simple nature of this Prime Ideal Principle enables us to generate a large number of hitherto unknown results of the “maximal implies prime” variety. The key notions used in our uniform approach to such prime ideal problems are those of Oka families and Ako families of ideals in a commutative ring, defined in (2.1) and (2.2). Much of this work has also natural interpretations in terms of categories of cyclic modules.     

Direct summands of direct sums of modules whose endomorphism rings have two maximal right ideals

Let M₁,…,M_n be right modules over a ring R. Suppose that the endomorphism ring of each module M_i has at most two maximal right ideals. Is it true that every direct summand of M₁⊕?⊕M_n is a direct sum of modules whose endomorphism rings also have at most two maximal right ideals? We show that the answer is negative in general, but affirmative under further hypotheses. The endomorphism ring of uniserial modules, that is, the modules whose lattice of submodules is linearly ordered under inclusion, always has at most two maximal right ideals, and Pavel P?íhoda showed in 2004 that the answer to our question is affirmative for direct sums of finitely many uniserial modules.