Generalized Affine Transformation Monoid of a Residue Class Ring

In this paper, we consider the generalized affine transformation monoid Gaff (A/I) of a residue class ring A/I of a commutative ring A with identity modulo its nonzero ideal I. For the general case, we investigate the Green's relations, Schutzenberger group for every D-class and, the structure of group H-classes, regular D-classes, the idempotent set E(Gaff (A/I)) and the regular element set Reg (Gaff (A/I)) of Gaff (A/I). If A is an integral domain and I a product of powers of invertible maximal ideals, we show that Gaff (A/I) is an epigroup, every R^⋆-classes of Gaff (A/I) is a nil-extension of a right group and that Gaff (A/I) is a complete lattice of nil-extensions of right groups.     

Exponential epigroups

We show that an exponential epigroup is a band of unipotent exponential epigroups (in which the set of idempotents forms a subsemigroup). Also, we investigate the structure of unipotent exponential epigroups. In addition, a characterization of the least normal band of groups congruence is given for an arbitrary medial epigroup. It is also shown that k-exponential eventually regular semigroups are necessarily epigroups.     

Permutative k-exponential epigroups

We prove that any permutative k-exponential epigroup is a normal band of unipotent epigroups. Moreover, we describe the least regular congruence on such semigroups. Finally, we give an example of a permutative epigroup which is not a band of unipotent epigroups.     

Algèbres amassées et algèbres préprojectives : le cas non simplement lacé

We generalize to the non simply-laced case results of Geiß, Leclerc and Schröer about the cluster structure of the coordinate ring of the maximal unipotent subgroups of simple Lie groups. In this way, cluster structures in the non simply-laced case can be seen as projections of cluster structures in the simply-laced case. This allows us to prove that cluster monomials are linearly independent in the non simply-laced case. To cite this article: L. Demonet, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Groupes libres et algebses de groupes en geometrie algebrique

Let X=Sp A be an affine scheme over a perfect field k of positive caracteristic. The functor D(k)-mod (W(A), (w(?)) is the unipotent part of the affine commutative free group generated by X (W is the ring of Witt-vectors and D the ring of Dieudonné). If X is a monoïd-scheme, this free group has a structure of ring scheme. In particular, if X=0 _k ^x , we obtain a new proof of the theorem of decomposition of the universal ring.     

On Maximal Unipotent Subgroups of a Special Linear Group Over Commutative Ring

Ukrainian Mathematical Journal - It is proved that all maximal unipotent subgroups of a special linear group over commutative ring with identity (such that the factor ring of its modulo primitive...     

Axiomatic definition of a quotient ring

The quotient ring QΦ(R) of the ring R with respect to a strong filter Φ of right ideals of R is axiomatically defined as a maximal strong Φ-essential extension of R. The ring QΦ(R) is constructively obtained in the form of lim Hom_r (I, R), where I ∈ Φ.     

ON ORDER RINGS OF SEMI-PRIMARY RINGS

The main results of this paper are stated as follows.Let R be an orderring in thesemi-primary ring Q.Suppose that R satisfies the maximal condition for nil right ideals ofR,Then we have(i)if an ideal I of R has a finite length as right R-module,then I alsohas a finite length as left R-module;(ii)denote by A(R)the Artinian radical of R,andN the nil radical of R,then A(R)+N/N=A(R/N),if R satisfies the commutative condi-tion on the zero product of prime ideals of B.     

Rigid modules over preprojective algebras

Let Λ be a preprojective algebra of simply laced Dynkin type Δ. We study maximal rigid Λ-modules, their endomorphism algebras and a mutation operation on these modules. This leads to a representation-theoretic construction of the cluster algebra structure on the ring ℂ[N] of polynomial functions on a maximal unipotent subgroup N of a complex Lie group of type Δ. As an application we obtain that all cluster monomials of ℂ[N] belong to the dual semicanonical basis. Mathematics Subject Classification (2000) 14M99, 16D70, 16E20, 16G20, 16G70, 17B37, 20G42     

Cancellable Elements of the Lattice of Epigroup Varieties

We completely describe all commutative epigroup varieties that are cancellable elements of the lattice EPI of all epigroup varieties. In particular, we prove that a commutative epigroup variety is a cancellable element of the lattice EPI if and only if it is a modular element of this lattice.     

A projective analogue of schur's tensor power construction

For a prime ring R with σ ? Aut(R), we examine the extended center of the skew power series ring R[[x;σ]]. We prove the extended center of R[[x;σ]] is isomorphic to: (1) C^σ if no power of σ is X-inner where C^σ is the fixed field of a acting on the extended center C of R (2) C^σ((vx')) (Laurent series) if some positive power of a is X-inner and R is closed prime. In this case I is the least positive integer such that a¹ is X-inner with σ¹= inn(v). We provide an example showing this result fails if R is not closed prime.     

Cohomology for infinitesimal unipotent algebraic and quantum groups

In this paper we study the structure of cohomology spaces for the Frobenius kernels of unipotent and parabolic algebraic group schemes and of their quantum analogs. Given a simple algebraic group G, a parabolic subgroup P _J , and its unipotent radical U _J , we determine the ring structure of the cohomology ring H^?((U _J )₁, k). We also obtain new results on computing H^?((P _J )₁, L(??)) as an L _J -module where L(??) is a simple G-module with highest weight ?? in the closure of the bottom p-alcove. Finally, we provide generalizations of all our results to small quantum groups at a root of unity.     

On divided commutative rings

Let R be a commutative ring with identity having total quotient ring T. A prime ideal P of R is called divided if P is comparable to every principal ideal of R. If every prime ideal of R is divided, then R is called a divided ring. If P is a nonprincipal divided prime, then P^-1 = { x ? T : xP ? P} is a ring. We show that if R is an atomic domain and divided, then the Krull dimension of R ≤ 1. Also, we show that if a finitely generated prime ideal containing a nonzerodivisor of a ring R is divided, then it is maximal and R is quasilocal.     

Large Abelian Unipotent Subgroups of Finite Chevalley Groups

We compute maximal orders of unipotent Abelian subgroups, estimate p-ranks, and describe the structure of Thompson subgroups of maximal unipotent subgroups of finite exceptional groups of Lie type.     

广义导子幂中心化子条件(英文)

设R是素环,U是R的极大右商环,C是R的扩展型心,I是R的非零理想,δ是R上的广义导子,a,b,q∈U.本文给出了恒等式(aδ(qx)-bx)~n=0(∈C在I上成立的充分必要条件,得到的结果推广了Herstein,Lanski,Lee的结论.     

On nonsingular right fpf rings

A right FPF ring is one over which every finitely generated faithful right module is a generator. The main purpose of the article is to givp the following cnaracterization of certain right FPF rings. TheoremLet R be semiprime and right semihereditary. Then R is right FPF iff (1) the right maximal ring of quotients Q_r (R) = Q coincides with the left and right classical rings of quotients and is self-injective regular of bounded index, (ii) R and Q have the same central idem-potents, (iii) if I is an ideal of R generated by a ma­ximal ideal of the boolean algebra of central idempotent s5 R/I is such that each non-zero finitely generated right ideal is a generator (hence prime), and (iv) R is such that every essential right ideal contains an ideal which is essential as a right ideal

In case that R is semiprime and module finite over its centre C, then the above can be used to show that R is FPF (both sides) if and only if it is a semi-hereditary maximal C-order in a self-injective regular ring (of finite index)

In order to prove the above it is shown that for any semiprime right FPF ring R, Q _lcl (R) exists and coincides with Q_r(R) (Faith and Page have shown that the latter is self-injective regular of bounded index). It R is semiprime right FPF and satisfies a polynamical identity then the factor rings as in (iii) above are right FPF and R is the ring of sections of a sheaf of prime right FPF rings

The Proofs use many results of C. Faith and S Page as well as some of the techniques of Pierce sheaves     

Invariant polynomials under the maximal unipotent subgroup of a classical group

Let N be a maximal unipotent subgroup of a classical complex Lie group G, whose Lie algebra we denote by g. Inside the ring of N-invariant polynomials S′(g)^N we consider the subring generated by the polynomials of weight e^{m(Λ + Λ*)}, where Λ and Λ^* are the highest weights corresponding to the natural representation of g and to its dual respectively, and m ε Z.We prove that this subring is a polynomial ring and we explicitly give a set of generators.     

LENGTH ONE IDEAL EXTENSIONS AND THEIR ASSOCIATED GRADED RINGS

Let (R. m) be a d-dimensional Cohen-Macaulay local ring. Given m-primary ideals J ? I of R such that I is contained in the integral closure of J and λ(I/J)= I, we compare depth G(J) and depth G(J). For example, if J has reduction number one, JI = I², and μ(J)≤ d + 1, we prove that depth G(I)≥d – 1. If, in addition, μ(I)= d + 1, we show that I has reduction number one, and hence G(I) is Cohen-Macaulay. These results, besides leading to statements comparing depths of associated graded rings along a composition series, make visible the possibility of studying powers of an ideal by using reductions that are not minimal reductions.     

关于单位正则环

本文得到了单位正则环的一个新特征,证明了:正则环R为单位正则环当且仅当存在理想I使得(1)R/I为单位正则环;(2)对任何a∈R,存在理想J满足JI=0和a=aua,其中u模J左可逆.作为应用,利用零化子理想刻画了单位正则环.     

Separative Ideals,Clean Elements,and Unit-Regularity

ABSTRACT

We prove that an ideal I of a regular ring R is separative if and only if each a ? R satisfying Rr(a)aR = Ra?(a)R = RaR(1 ? a)R ? I is unit-regular. If I is a separative ideal of a regular ring R, then each a ? R satisfying Rar(a²) = ?(a²)aR = R(a ? a²) R ? I is clean. Some applications are also obtained.