Invariants of Lie superalgebras acting on associative algebras

LetR be a semiprime algebra over a fieldK acted on by a finite-dimensional Lie superalgebraL. The purpose of this paper is to prove a series of going-up results showing how the structure of the subalgebra of invariantsR ^Lis related to that ofR. Combining several of our main results we have: Theorem: Let R be a semiprime K-algebra acted on by a finite-dimensional nilpotent Lie superalgebra L such that if characteristic K=p then L is restricted and if characteristic, K=0 then L acts on R as algebraic derivations and algebraic superderivations.

1. If R^L is right Noetherian, then R is a Noetherian right R^L-module. In particular, R is right Noetherian and is a finitely generated right R^L-module.
2. If R^L is right Artinian, then R is an Artinian right R^L-module. In particular, R is right Artinian and is a finitely generated right R^L-module.
3. If R^L is finite-dimensional over K then R is also finite-dimensional over K.
4. If R^L has finite Goldie dimension as a right R^L-module, then R has finite Goldie dimension as a right R-module.
5. If R^L has Krull dimension α as a right R^L-module, then R has Krull dimension α as a right R^L-module. Thus R has Krull dimension at most α as a right R-module.
6. If R is prime and R^L is central, then R satisfies a polynomial identity.
7. If L is a Lie algebra and R^L is central, then R satisfies a polynomial identity.

We also provide counterexamples to many questions which arise in view of the results in this paper.     

The connes spectrums of certain finite non-abelian groups

In this article, we investigate when every simple module has a projective (pre)envelope. It is proven that (1) every simple right R-module has a projective preenvelope if and only if the left annihilator of every maximal right ideal of R is finitely generated; (2) every simple right R-module has an epic projective envelope if and only if R is a right PS ring; (3) Every simple right R-module has a monic projective preenvelope if and only if R is a right Kasch ring and the left annihilator of every maximal right ideal of R is finitely generated.     

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     

Chain conditions in modules with krull dimension

Any ring with Krull dimension satisfies the ascending chain condition on semiprime ideals. This result does not hold more generally for modules. In particular if Ris the first Weyl algebra over a field of characteristic 0 then there are Artinian R-modules which do not satisfy the ascending chain condition on prime submodules. However, if Ris a ring which satisfies a polynomial identity then any R-module with Krull dimension satisfies the ascending chain condition on prime submodules, and, if Ris left Noethe-rian, also the ascending chain condition on semiprime submodules.     

左连续环中若干链条件的等价性

(1)设R是左连续环,则R是左Artin环当且仅当R满足左限制有限条件当且仅当R关于本质左理想满足极小条件当且仅当R关于本质左理想满足极大条件.同时给出一个左自内射环是QF环的充要条件;(2)证明了左Z₁-环上的有限生成模都有Artin-Rees性质.     

Some examples in PI ring theory

Several examples are constructed, including a finite ring which cannot be embedded in matrices over any commutative ring, a semiprime PI ring with no classical ring of quotients, an example showing that the property of having all regular elements invertible is not inherited by matrix ringsM _n(R), and a prime PI ringR with an idempotente such thatR/ReR has finitely generated projective modules not induced by any finitely-generated projective R-module. Most of this work was done while the author was a guest of the University of Leeds’ Ring Theory Year (1972–1973), with the support of an Alfred P. Sloan Fellowship, and under the stimulating influence of Lance W. Small.     

The Krull dimension of the module category over right noetherian serial rings

For a right Noetherian serial ring R that is not Artinian, it is proved that the Krull dimension of the category of finitely generated right R-modules is equal to one. Bibliography: 17titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 73–86.     

Algebraic prime subalgebras in simple Artinian algebras

We prove a Wedderburn-Artin type theorem for algebraic prime subalgebras in simple Artinian algebras, giving a generalized version of Yahaghi’s theorem [B.R. Yahaghi, On F-algebras of algebraic matrices over a subfield F of the center of a division ring, Linear Algebra Appl. 418 (2006) 599-613]. We also show that every semiprime left algebraic subring in a semiprime right Goldie ring must be a semiprime Artinian ring.     

ARTINIAN RADICAL AND ITS APPLICATION

Let R be a left and right Neotherian ring with identity. Let A be the Artinian radical.Lenagan [3]pointed out that R has Artinian quotient ring if A=O and the Krull dimensionof R is one. In this paper first the structure of Artinian radical is investigated. Then forR with Krull dimension one the author gives a necessary and sufficient condition under whichR has Artinian quotient ring. The main results are as follows: (i) A=eR, where e is acentral idempotent element of R, if and only if r(A)~λ=l(A)~λ=(∩ i=1 ,…, n_k k=1, …, p(a_~(k)))~λ, where λ isa positive integer, p(a_i~(k)) are prime ideals of i~ and ~(A)(lA)) is the notation of right(:eft)annihilatorof A(see Theorem 7). (ii) In the case(i)R=A~η(A)~λ. (iii) If R has Krulldimension one, then R has Artinian quotient ring if and only if there exists a positivointeger λ such that (A)~λ=l(A)~λ=(∩ i=1 ,…, n_k k=1, …, p(a_~(k)))~λ.     

Complete flat modules

Let R be a commutative and noetherian ring. It is known tht if R is local with maximal ideal M and F is a flat R-module, then the Hausdorff completion F of F with the M-adic topology is flat. We show that if we assume that the Krull dimension of R is finite, then for any ideal I C R, the Hausdorff completion F* of a flat module F with the I-adic topology is flat. Furthermore, for a flat module F over such R, there is a largest ideal I such that F is Hausdorff and complete with the I-adic topology. For this I, the flat R/I-module F/IF will not be Hausdorff and complete with respect to the topology defined by any non-zero ideal of R/I. As a tool in proving the above, we will show that when R has finite Krull dimension, the I-adic Hausdorff completion of a minimal pure injective resolution of a flat module F is a minimal pure injective resolution of its completion F*. Then it will be shown that flat modules behave like finitely generated modules in the sense that on F* the I-adic and the completion topologies coincide, so F* is I-adically complete.     

广义FP—内射模、广义平坦模与某些环

左（右）R－模A称为GFP－内射模，如果ExtR(M,A)=0对任－2－表现R－模M成立；左（右）R－模称为G－平坦的，如果Tor1^R(M,A)=0(Tor1^R(AM)=0)对于任一2－表现右（左）R－模M成立；环R称左（右）R－半遗传环，如果投射左（右）R－模的有限表现子模是投射的，环R称为左（右）G－正而环，如果自由左（右）R－模的有限表现子模为其直和项，研究了GFP－内射模和G－平坦模的一些性质，给出了它们的一些等价刻划，并利用它们刻划了凝聚环，G－半遗传环和G－正则环。     

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.     

On chains of prime submodules

In this paper, we study the dimension of a module over a commutative ring, which is defined to be the length of a longest chain of prime submodules. This notion is analogous to the usual Krull dimension of a ring. We investigate how some bounds on the dimension of modules are related to the structure of the underlying ring. The dimension of finitely generated modules over a Dedekind domain is also studied. By examining the structure of prime submodules, a formula for the dimension of a free module of finite rank, over a Noetherian one-dimensional domain, is obtained.     

On the exactness of flat resolvents

Let R be a left coherent ring, FP — idRR the FP — injective dimension of RR and wD(R) the weak global dimension of R. It is shown that 1) FP -idRR < n ( n > 0) if and only if every flat resolvent 0 → M → F° → F¹... of a finitely presented right R—module M is exact at F'(i > n?1) if and only if every nth F -cosyzygy of a finitely presented right R — module has a flat preenvelope which is a monomorphism; 2) wD(R) < n (n > 1) if and only if every (n?l)th F-cosyzygy of a finitely presented right R—module has a flat preenvelope which is an epimorphism; 3) wD(R) 0) if and only if every nth F — cosyzygy of a finitely presented right R — module is flat. In particular, left FC rings and left semihereditary rings are characterized     

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.     

FGF group rings

Periodica Mathematica Hungarica - A ring R is called a left FGF ring if every finitely generated left R-module can be embedded in a free left R-module. It is proved that a group ring RG is left FGF...     

Ass（M／N） and Ann（M／N） on the Noetherian Module M

Let R is a noetherian ring,M is a finitely generated R-module.This paper studies the relationbetween associated prime Ass(M/N)and annihilator Ann(M/N),and has given the necessary andsufficient conditions of Ass(M/N)=Ann(M/N).     

关于拟P-内射模的一些注记(英文)

设 R是一个环 .一个右 R-模 M叫做拟 P-内射的 ,如果 M的每个 M-循环子模到 M的任一个 R-同态都能扩展到 M.假设 M是一个自生成子的拟 P-内射模 .在这篇文章中 ,我们表明如果这样一个模是一个 CF-模 (特别地 ,CS-模 ) ,那么 S/J(S)是正则的 ,其中 S=End(MR) .进一步 ,如果 S是半素环 ,那么 M的每个极大核是 M的一个直和项 .这些结果扩展了 P-内射环的一些结果