本原环的Grothendieck群

设R为本原环,对应的忠实既约模为T,且soc(R)≠R,设R=R/soc(R).在文中证明了以下结果: (1)K_o(R)→K_o(R)是满同态,且当soc(R)≠0时,N=Ker(K_o(R)→K_o(R))是由[T]∈K_o(R)生成的循环子群. (2)若soc(R)=0,则存在一个本原环R_1,soc(R_1)≠0,使得R是R_1的同态象,且K_o(R_1)≌K_o(R)⊕N,其中N=Ker(K_o(R_1)→K_o(R))是由[T]∈K_o(R_t)生成的循环子群.     

一类阿基米德半群的构造及其同余格

本文引入同底的π-左、右零半群的夹群积并用来刻划带本原幂等元的阿基米德半群的构造.文中讨论了有限阶阿基米德半群的同余格,并证明了当有限阶阿基米德半群的正则R,L类的个数不超过5时,它的同余格是半模格.     

环上群环的半单性——关于G.Connell的一个猜测

设环R有1,G是群。用R(G)表示R、G的群环。0(G)表示群G子群的阶的集合。任意域F(ch.F0(G))上群环F(G)的J一半单性问题,至今仅证明对某些群,如局部有限群、局部可解群、Abel群、有序群等时R(G)是半本原环。G.Connell于63年将域扩展到环,他得出当环R可换时,R(G)是半素与半本原的充要条件([2]定理5、6),并断言要去掉R的可换条件是很困难的,但他猜测前者R的可换条件有可能去掉。     

关于适中稳定环

引入了一类新的稳定环--适中稳定环,证明了如果R是Artin本原因子的置换环,则R是适中稳定环当且仅当1R=α+β,其中α,β∈U(R).如果R是适中稳定环,则K1(R)(≌)U(R)/θ(R),这里θ(R)是V(R)与U(R)'之间的适中子群.作为应用,计算了Artin本原因子的置换环的K1群.     

关于有限生成投射模的进一步结果

本文研究所有右本原同态象均为阿丁环的调换环,刻画了其K0-群并证明了在有限生成投射模的范畴中关于直和的在同构意义下的n-th root总是唯一的.     

关于半完全环的K1群

本文研究了半完全环的K1群. 利用半局部环的K1群的已知结果和半完全环的构造, 证明了半完全环R的K1群是R的一些子环的K1群的直和与R的单位群的一个子群的商群.     

有限局部环上酉群阶的计算

设K=F_(q^2),其特征为p, q=p^α,K有对合自同构ω:a→a^q. G是一个p 群，其阶为p^β, 群代数R=KG为一局部环. K的2阶自同构ω可延拓为R的一个2阶自同构，记为ω'，为方便，对任意a∈R, 记ω‘(a)为~a. R上2n级酉群定义为U_(2n)R={A∈GL_(2n)R|A(0,I^n,I^n,0)~A^t=(0,I^n,I^n,0)} 该文计算了U_(2n)R的阶.      

Noether环上的幂稳定自由模

设I是Noether环R的投射理想, I^m=Iⁿ, m≠n. 该文证明, 有限生成投射右R - 模幂稳定自由当且仅当(1) 存在环S使得I^|m-n|( S ( R且有限生成投射S - 模是幂稳定自由; (2) 有限生成投射右R/I^|m-n| - 模幂稳定自由.     

矩阵环的有限乘法子群

将Minkowski关于有限整数矩阵群的著名结果推广到一般的环上,主要结果是证明了:对任意环R,如果R的加法群为有限生成的自由Abel群,则R的所有乘法可逆元构成的群U(R)中的有限子群精确到同构只有有限多个.     

环的广义斜导子

设R是一个半素环, RF(resp．Q)是它的左Martindale商环(对称Martindale 商环),K是R的一个本质理想,则K上的每一个广义斜导子μ能被唯一地扩展到RF和Q 上．设R是一个素环,K是R的一个本质理想,μ是K上的一个广义斜导子且α为其伴随自同构,d为其伴随斜导子,如果存在n≥0,使得对任意的x∈K都有μ(x)n=0,那么μ=0．     

替换环什么时候是冯诺依曼正则环（英文）

We study when exchange rings are von Neumann regular. An exchange ring R with primitive factors Artinian is von Neumann regular, if the Jacobson radical of any indecomposable homomorphic image of R is T-nilpotent, and if any indecomposable homomorphic image of R is semiprime. Every indecomposable semiprimitive factor ring of R is regular, if R is an exchange ring such that every left primitive factor ring of R is a ring of index at most n and if R has nil-property.     

ON EXCHANGE RINGS WITH BOUNDED INDEX OF NILPOTENCE

This paper studies exchange rings R such that R/J(R) has bounded index of nilpotence. We give several characterizations of such rings. We prove that if a semiprimitive exchange ring R has index n, then for any maximal two-sided I of R, if R/I has length n, then there exists a central idempotent element e in R such that eRe is an n by n full matrix ring over some exchange ring with central idempotents, and the restriction π from eRe to R/I is surjective.     

On commutative rings which are strongly prüfer

Abstract

As defined by Nicholson [Nicholson, W. K. (1977). Lifting idempotents and exchange rings. Trans. Amer. Math. Soc. 229:269–278] an element of a ring R is clean if it is the sum of a unit and an idempotent, and a subset A of R is clean if every element of A is clean. It is shown that a semiprimitive Gelfand ring R is clean if and only if Max(R) is zero-dimensional; if and only if for each M ∈ Max(R), the intersection all prime ideals contained in M is generated by a set of idempotents. We also give several equivalent conditions for clean functional rings. In fact, a functional ring R is clean if and only if the set of clean elements is closed under sum; if and only if every zero-divisor is clean; if and only if; R has a clean prime ideal.     

环的投射生成元和K0群

设Ｒ是含幺结合环,Ｐｇ（Ｒ）是Ｒ的所有投射生成元的同构类组成的半群,Ｇｒ（Ｐｇ（Ｒ））是Ｐｇ（Ｒ）的Ｇｒｏｔｈｅｎｄｉｅｃｋ群,在本文中我们证明了Ｋ０（Ｒ）＝Ｇｒ（Ｐｇ（Ｒ））。由此我们得到对任意ＶＢＮ环Ｒ,存在环Ｓ满足Ｓ＾２＝Ｓ并且具有Ａｕｔ－Ｐｉｃ性质,最后我们给出了环的一个分类,并且用Ｐｇ（Ｒ）的周期性对它作了描述。     

On N0-quasi-continuous exchange rings

An associative ring R with identity is said to have stable range one if for any a,b? R with aR + bR = R, there exists y ? R such that a + by is left (equivalently, right) invertible. The main results of this note are Theorem 2: A left or right continuous ring R has stable range one if and only if R is directly finite (i.e xy = 1 implies yx = 1 for all x,y ? R), Theorem 6: A left or right N ₀o-quasi-continuous exchange ring has stable range one if and only if it is directly finite, and Theorem 12: left or right N ₀-quasi-continuous strongly π-regular rings have stable range one. Theorem 6 generalizes a well-known result of Goodearl [10], which says that a directly finite, right N _o-continuous von Neumann regular ring is unit-regular     

Ideal-Symmetric and Semiprime Rings

Lambek extended the usual commutative ideal theory to ideals in noncommutative rings, calling an ideal A of a ring R symmetric if rst ∈ A implies rts ∈ A for r, s, t ∈ R. R is usually called symmetric if 0 is a symmetric ideal. This naturally gives rise to extending the study of symmetric ring property to the lattice of ideals. In the process, we introduce the concept of an ideal-symmetric ring. We first characterize the class of ideal-symmetric rings and show that this ideal-symmetric property is Morita invariant. We provide a method of constructing an ideal-symmetric ring (but not semiprime) from any given semiprime ring, noting that semiprime rings are ideal-symmetric. We investigate the structure of minimal ideal-symmetric rings completely, finding two kinds of basic forms of finite ideal-symmetric rings. It is also shown that the ideal-symmetric property can go up to right quotient rings in relation with regular elements. The polynomial ring R[x] over an ideal-symmetric ring R need not be ideal-symmetric, but it is shown that the factor ring R[x]/xⁿR[x] is ideal-symmetric over a semiprime ring R.     

Semiprimitivity of Orthodox Semigroup Algebras

Let S be a finite orthodox semigroup or an orthodox semigroup where the idempotent band E(S) is locally pseudofinite. In this paper, by using principal factors and Rukolaǐne idempotents, we show that the contracted semigroup algebra R₀[S] is semiprimitive if and only if S is an inverse semigroup and R[G] is semiprimitive for each maximal subgroup G of S. This theorem strengthens previous results about the semiprimitivity of inverse semigroup algebras.     

On the Semiprimitivity of Skew Polynomial Rings

Let R be a semiprime left Goldie ring with a monomorphism and an α-derivation, then is semiprimitive left Goldie ring.