Chinese Algebras of Rank 3

The structure of the algebra K[M] of the Chinese monoid M of rank 3 over a field K is studied. The minimal prime ideals are described and the classical Krull dimension is computed. It follows that every minimal prime ideal is determined by a homogeneous congruence on M. Moreover, the prime radical is nilpotent and equal to the Jacobson radical. This ideal is not determined by a congruence on M.     

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.     

Submonoids of Polycyclic-by-Finite Groups and their Algebras

We describe Noetherian semigroup algebras K[S] of submonoids S of polycyclic-by-finite groups over a field K. As an application, we show that these algebras are finitely presented and also that they are Jacobson rings. Next we show that every prime ideal P of K[S] is strongly related to a prime ideal of the group algebra of a subgroup of the quotient group of S via a generalised matrix ring structure on K[S]/P. Applications to the classical Krull dimension, prime spectrum, and irreducible K[S]-modules are given.     

Resolutions of subsets of finite sets of points in projective space

Abstract

Eisenbud et al. proved a number of results regarding Gröbner bases and initial ideals of those ideals J in the free associative algebra K ?X ₁,…, X _n ? which contain the commutator ideal. We prove similar results for ideals which contains the anti-commutator ideal (the defining ideal of the exterior algebra). We define one weak notion of generic initial ideals in K ?X ₁,…, X _n ?, and show that generic initial ideals of ideals containing the anti-commutator ideal, or the commutator ideal, are finitely generated.     

Plactic algebra of rank 3

The structure of the algebra K[M] of the plactic monoid M of rank 3 over a field K is studied. The minimal prime ideals of K[M] are described. There are only two such ideals and each of them is a principal ideal determined by a homogeneous congruence on M. Moreover, in case K is uncountable and algebraically closed, the left and right primitive spectrum and the corresponding irreducible representations of the algebra K[M] are described. All these representations are monomial. As an application, a new proof of the semiprimitivity of K[M] is given.     

Nilpotent Lie Algebras with a Small Second Derived Quotient 1

Here we find the structure of nilpotent Lie algebras L with dim(L′/L″) = 3 and L″ ≠ 0. Following the pattern of results of Csaba Schneider in p-groups, we show that L is the central direct sum of ideals H and U, where U is the direct sum of a generalized Heisenberg Lie algebra and an abelian Lie algebra. We then find over the complex numbers that H falls into one of fourteen isomorphism classes.     

A Note on Semigroup Algebras of Permutable Semigroups

Let S be a semigroup and 𝔽 be a field. For an ideal J of the semigroup algebra 𝔽[S] of S over 𝔽, let ?_J denote the restriction (to S) of the congruence on 𝔽[S] defined by the ideal J. A semigroup S is called a permutable semigroup if α ○ β = β ○ α is satisfied for all congruences α and β of S. In this paper we show that if S is a semilattice or a rectangular band then φ_{{S; 𝔽}}: J → ?_J is a homomorphism of the semigroup (Con(𝔽[S]); ○ ) into the relation semigroup (?_S; ○ ) if and only if S is a permutable semigroup.     

On Behavior of Solvable Ideals of Lie Algebras Under Outer Derivations

Let L be a finite dimensional Lie algebra over a field F. It is well known that the solvable radical S(L) of the algebra L is a characteristic ideal of L if char F = 0, and there are counterexamples to this statement in case char F = p > 0. We prove that the sum S(L) of all solvable ideals of a Lie algebra L (not necessarily finite dimensional) is a characteristic ideal of L in the following cases: 1) char F = 0; 2) S(L) is solvable and its derived length is less than log₂ p.     

Non-Unique Factorization of Polynomials Over Residue Class Rings of the Integers

We investigate non-unique factorization of polynomials in ? _{p ⁿ} [x] into irreducibles. As a Noetherian ring whose zero-divisors are contained in the Jacobson radical, ? _{p ⁿ} [x] is atomic. We reduce the question of factoring arbitrary nonzero polynomials into irreducibles to the problem of factoring monic polynomials into monic irreducibles. The multiplicative monoid of monic polynomials of ? _{p ⁿ} [x] is a direct sum of monoids corresponding to irreducible polynomials in ? _p [x], and we show that each of these monoids has infinite elasticity. Moreover, for every m ∈ ?, there exists in each of these monoids a product of 2 irreducibles that can also be represented as a product of m irreducibles.     

Classes of Graded Ideals with Given Data in the Exterior Algebra

Let ? be the family of graded ideals J in the exterior algebra E of a n-dimensional vector space over a field K such that e(E/J) = dim _K (E/J) = e, indeg(E/J) = i and H _E/J (i) = dim _K (E/J) _i are fixed integers. It is shown that there exists a unique lexsegment graded ideal J(n, e, i) ? ? whose Betti numbers give an upper bound for the Betti numbers of the ideals of ?. The authors continue the computation of upper bounds for the Betti numbers of graded ideals with given data started in Crupi and Utano (1999 Crupi , M. , Utano , R. ( 1999 ). Upper bounds for the Betti numbers of graded ideals of a given length in the exterior algebra . Comm. Alg. 27 : 4607 – 4631 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]).     

Properness,strictness, and nilness in jordan systems

The basic theme of this paper, suggested by Ottmar Loos, is to show that certain sets S ofelements in a Jordan system J form an ideal by showing that S = J∩R([Jtilde]) is the Amitsur shrinkage of some well-knownl radical R on some extension [Jtilde] of J. The Jacobson radical Rad(J) and the degenerate radical Deg(J) have elemental characterizations as (respectively) the properly quasi-invertible elements and the m-finite elements. Frojn these two characterizations we show that: (1) the strictly properly nilpotent elements coincide with the strictly properly quasi-invertible elements and form the ideal J∩Rad(J[T]) (2) the strictly m-finite elements coincide with the m-finite elements and form the ideal Deg:(J) (3) the m-bounded elements form an ideal J ∩ Deg(Seq(J)) (Seq the algebra of sequences); and (4) the strictly m-bounded elements coincide with the strictly properly nilpotence-bounded elements and form the ideal J ∩ Deg(Seq(J[t])). We show that all these constructions are stable under structural pairs, a useful generalization of the concept of structural transformation. The question of whether the properly nilpotent elements form an ideal, and if so whether this is the nil radical, is an open question intimately related to the Kothe Conjecture for associative algebras.     

Nilpotent n-Lie Algebras

We find examples of nilpotent n-Lie algebras and prove n-Lie analogs of classical group theory and Lie algebra results. As an example we show that a nilpotent ideal I of class c in a n-Lie algebra A with A/I ² nilpotent of class d is nilpotent and find a bound on the class of A. We also find that some classical group theory and Lie algebra results do not hold in n-Lie algebras. In particular, non-nilpotent n-Lie algebras can admit a regular automorphism of order p, and the sum of nilpotent ideals need not be nilpotent.     

Projective Summands in Tensor Products of Simple Modules of Finite Dimensional Hopf Algebras

Abstract

Let H be a finite dimensional Hopf algebra over a field k. We show that H contains a unique maximal Hopf ideal J _w (H) contained in J(H), the Jacobson radical of H. We give various characterizations of J _w (H), for example J _w (H) = Ann _H ((H/J(H))^?n ) for all large enough n. The smallest positive integer n with this property is denoted by l _w (H). We prove that l _w (H) equals the smallest number n such that (H/J(H))^?n contains every projective indecomposable H/J _w (H)-module as a direct summand. This also equals the minimal n such that the tensor product of n suitable simple H-modules contains the projective cover of the trivial H/J _w (H)-module as a direct summand. We define projective homomorphisms between H-modules, which are used to obtain various reciprocity laws for tensor products of simple H-modules and their projective indecomposable direct summands. We also discuss some consequences of our general results in case H = kG is a group algebra of a finite group G and k is a field of characteristic p.     

On Lie Nilpotent Rings and Cohen's Theorem

We study certain (two-sided) nil ideals and nilpotent ideals in a Lie nilpotent ring R. Our results lead us to showing that the prime radical rad(R) of R comprises the nilpotent elements of R, and that if L is a left ideal of R, then L + rad(R) is a two-sided ideal of R. This in turn leads to a Lie nilpotent version of Cohen's theorem, namely if R is a Lie nilpotent ring and every prime (two-sided) ideal of R is finitely generated as a left ideal, then every left ideal of R containing the prime radical of R is finitely generated (as a left ideal). For an arbitrary ring R with identity we also consider its so-called n-th Lie center Z _n (R), n ≥ 1, which is a Lie nilpotent ring of index n. We prove that if C is a commutative submonoid of the multiplicative monoid of R, then the subring ?Z _n (R) ∪ C? of R generated by the subset Z _n (R) ∪ C of R is also Lie nilpotent of index n.     

THE M-VALUATION SPECTRUM OF A COMMUTATIVE RING

Abstract

In 1945, N. Jacobson has introduced the definition of radical of a ring A (which is known as “Jacobson radical”, and is denoted J = J(A)). Later the concept of (Jacobson) radical of a left (or right) A-module M, J(M), has been defined as the intersection of all submodules N ≤ M such that M/N is simple. Thus one may consider the left radical J _l  = J( _A A) and the right radical J _r  = J(A _A ) of A, which are bilateral ideals of A, and are contained in J(A). If A has identity, one has J = J _l  = J _r , but this equality is not valid in general. Dual, it is possible to define left socle S _l and right socle S _r of A. We shall establish relations between J, J _l , J _r , S _l and S _r , and for artinian algebras we shall obtain expressions for J _l (A) and J _r (A), S _l (A) and S _r (A). In particular, if A is a finite dimensional algebra over a field we display J _l  = J( _A A) (and J _r  ? J(A _A )) in a matrix representation.     

Efficient Generation of Prime Ideals in Polynomial Rings up to Radical

We call an ideal I of a ring R radically perfect if among all ideals whose radical is equal to the radical of I, the one with the least number of generators has this number of generators equal to the height of I. Let R be a ring and R[X] be the polynomial ring over R. We prove that if R is a strong S-domain of finite Krull dimension and if each nonzero element of R is contained in finitely many maximal ideals of R, then each maximal ideal of R[X] of maximal height is the J _max-radical of an ideal generated by two elements. We also show that if R is a Prüfer domain of finite Krull dimension with coprimely packed set of maximal ideals, then for each maximal ideal M of R, the prime ideal MR[X] of R[X] is radically perfect if and only if R is of dimension one and each maximal ideal of R is the radical of a principal ideal. We then prove that the above conditions on the Prüfer domain R also imply that a power of each finitely generated maximal ideal of R is principal. This result naturally raises the question whether the same conditions on R imply that the Picard group of R is torsion, and we prove this to be so when either R is an almost Dedekind domain or a Prüfer domain with an extra condition imposed on it.     

Serre's Condition Rℓ for Affine Semigroup Rings

In this note we characterize the affine semigroup rings K[S] over an arbitrary field K that satisfy condition R_? of Serre. Our characterization is in terms of the face lattice of the positive cone pos(S) of S. We start by reviewing some basic facts about the faces of pos(S) and consequences for the monomial primes of K[S]. After proving our characterization we turn our attention to the Rees algebras of a special class of monomial ideals in a polynomial ring over a field. In this special case, some of the characterizing criteria are always satisfied. We give examples of non-normal affine semigroup rings that satisfy R₂.     

Semigroup Algebras and Noetherian Maximal Orders: a Survey

A survey is given on recent results describing when a semigroup algebra K[S] of a submonoid S of a polycyclic-by-finite group is a prime Noetherian maximal order. As an application one constructs concrete classes of finitely presented algebras that have the listed properties. Also some open problems are stated.