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1.
《代数通讯》2013,41(12):5701-5715
We investigate when semigroup algebras K[S] of submonoids S of torsion free polycyclic-by-finite groups G are Noetherian unique factorization rings in the sense of Chatters and Jordan, that is, every prime ideal contains a principal height one prime ideal. For the group algebra K[G] this problem was solved by Brown.  相似文献   

2.
《代数通讯》2013,41(7):2733-2742
Abstract

For a class of prime ideals P of a cancellative semigroup S it is shown that the factors S/P have a structure of a monomial semigroup over a group. Consequences for the semigroup algebras K[S] are discussed.  相似文献   

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.  相似文献   

4.
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.  相似文献   

5.
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.  相似文献   

6.
Jan Krempa 《代数通讯》2013,41(1):98-103
We construct a finitely generated monoid S with a zero element such that for every field K the Jacobson radical of the monoid algebra K[S] is a sum of nilpotent ideals but is not nilpotent. Moreover, the contracted monoid algebra K 0[S] is a monomial algebra.

If K is a field of characteristic p > 0, then we construct a finitely presented group H p such that the Jacobson radical J of the group algebra K[H p ] is a sum of nilpotent ideals, but is not nilpotent. Moreover, K[H p ]/J is a domain.  相似文献   

7.
Simple algebras of Weyl type   总被引:9,自引:0,他引:9  
Over a fieldF of arbitrary characteristic, we define the associative and the Lie algebras of Weyl type on the same vector spaceA[D] =AF[D] from any pair of a commutative associative algebra,A with an identity element and the polynomial algebraF[D] of a commutative derivation subalgebraD ofA We prove thatA[D], as a Lie algebra (modulo its center) or as an associative algebra, is simple if and only ifA isD-simple andA[D] acts faithfully onA. Thus we obtain a lot of simple algebras. Su, Y., Zhao, K., Second cohornology group of generalized Witt type Lie algebras and certain representations, submitted to publication  相似文献   

8.
An associative algebra R over a field K is said to be right ?-prime if for every nonzero r ? R, there exists a finitely generated subalgebra S of R such that rSt = 0 implies t = 0. Clearly, strongly prime implies ?-prime and ?-prime implies prime. A large number of examples of group algebras are given which show that the concept of ?-prime lies strictly between prime and strongly prime. A complete characterization of ?-prime group algebras is given. It is proved that a group algebra KG of the group G over the field K is ?-prime if and only if Λ+(G) = (1). Intersection theorems play an important role in the study. In the process, a new intersection theorem for ?-prime group algebras is obtained. Elementwise characterization of the ?-prime radical is given and its relation with some well-known radicals is discussed.  相似文献   

9.
A. Nagy  M. Zubor 《代数通讯》2013,41(11):4865-4873
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.  相似文献   

10.
Necessary and sufficient conditions are given for a prime Noetherian algebra K[S] of a submonoid S of a polycyclic-by-finite group G to be a maximal order. These conditions are entirely in terms of the monoid S. This extends earlier results of Brown concerned with the group ring case and of the authors for the case where K[S] satisfies a polynomial identity.  相似文献   

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