幺半群上的强α-自反环（英文）

For a monoid M and an endomorphism α of a ring R, we introduce the notion of strongly M-α-reflexive rings and study its properties. For an u.p.-monoid M and a right Ore ring R with its classical right quotient ring Q, we prove that R is strongly M-α-reflexive if and only if Q is strongly M-α-reflexive, where R is α-rigid, α is an epimorphism of R. The relationship between some special subrings of upper triangular matrix rings and strongly M-α-reflexive rings is also investigated. Several known results similar to strongly M-α-reversible rings are obtained.     

幺半群上的斜强可逆环

For a monoid M, we introduce the concept of skew strongly M-reversible rings which is a generalization of strongly M-reversible rings, and investigate their properties. It is shown that if G is a finitely generated Abelian group, then G is torsion-free if and only if there exists a ring R with |R| ≥ 2 such that R is skew strongly G-reversible. Moreover, we prove that if R is a right Ore ring with classical right quotient ring Q, then R is skew strongly M-reversible if and only if Q is skew strongly M-reversible.     

关于Noetherian环上拓扑滤子交换性的一个注记

Let R be a ring with identity,and R-fil denote the set of all left topolo-gizing filters on R.In this paper,we give a sufficient condition for the commutativityof R-fil under the hypothesis of left Noetherianness,as well as some examples.     

Rectangular Ring Congruences on an E-inversive Semiring

In this paper, we discussed the property of rectangular band semiring congruence and ring congruence on a semiring and gave some characterizations and structure of rectangular ring congruence on an E-inversive semiring.     

代数整数环的商环

In this paper,we discuss on the algebraic interger ring of the quadratic field of Q.The form of the quotient ring of the algebraic interger ring with respect to the order ideal is presented.     

具有对称自同态的环及其扩张

Let R be a ring with an endomorphism α and an α-derivation δ. We introduce the notions of symmetric α-rings and weak symmetric α-rings which are generalizations of symmetric rings and weak symmetric rings, respectively, discuss the relations between symmetricα-rings and related rings and investigate their extensions. We prove that if R is a reduced ring and α(1) = 1, then R is a symmetric α-ring if and only if R[x]/(x n) is a symmetric ˉα-ring for any positive integer n. Moreover, it is proven that if R is a right Ore ring, α an automorphism of R and Q(R) the classical right quotient ring of R, then R is a symmetric α-ring if and only if Q(R) is a symmetric ˉα-ring. Among others we also show that if a ring R is weakly 2-primal and(α, δ)-compatible, then R is a weak symmetric α-ring if and only if the Ore extension R[x; α, δ] of R is a weak symmetric ˉα-ring.     

正则环和GP-内射环

A ring R is called left Gp-injective if for any a∈R, there exists a positive integer n such that any left R-homomorphism of Raⁿ into R extends to one of R into R. In this paper, we prove that the centre of semiprime (left nonsingular) left GP- injective ring is regular ring, and improve some propositions in [3].     

关于拟赋值环的一个刻画

Let R be a commutative ring without nil-factor. In this paper, we discuss the problem of quasi-valuation ring presented in the reference “Wang Shianghaw, On quasi-valuation ring, Northeast People‘s Univ. Natur. Sci. J., (1)(1957), 27-40”,when the quotient field of R is an algebraic number field or an algebraic function field, and we obtain a characterization of quasi-valuation rings.     

Principal quasi-Baerness of formal power series rings

Let R be a ring. We consider left （or right） principal quasi-Baerness of the left skew formal power series ring R[[x;α]] over R where a is a ring automorphism of R. We give a necessary and sufficient condition under which the ring R[[x; α]] is left （or right） principally quasi-Baer. As an application we show that R[[x]] is left principally quasi-Baer if and only if R is left principally quasi- Baer and the left annihilator of the left ideal generated by any countable family of idempotents in R is generated by an idempotent.     

T正则环与TV环

In this paper, we generalize flat modules, regular rings and V- rings to the situation of a hereditary torsion theory, that a ring R is regular if and only if R is a left nonsingular ring and for every essential left ideal o of R, and for eve ry element a∈o there exists a element a′∈o , such that a= aa′. (theorem 3.3).     

Generalized Differential Identities of （Semi-）Prime Rings

Let R be a semiprime ring with characteristic p≥0 and RF be its left Martindale quotient ring. If ф（Xi^△j） is a reduced generalized differential identity for an essential ideal of R, then ф（Zije（△j ）） is a generalized polynomial identity for RF, where e（△j） are idempotents in the extended centroid of R determined by △j. Let R be a prime ring and Q be its symmetric Martindale quotient ring. If ф（Xi△j） is a reduced generalized differential identity for a noncommutative Lie ideal of R, then ф（Zij） is a generalized polynomial identity for [R, R]. Moreover, if ф（Xi△j） is a reduced generalized differential identity, with coefficients in Q, for a large right ideal of R, then ф（Zij） is a generalized polynomial identity for Q.     

Generalized Smash Products

In this paper．we study the ring #(D．B)and obtain two very interesting results. First we prove in Theorem 3 that the category of rational left BU-modules is equivalent to both the category of #-rational left modules and the category of all(B．D)-Hopf modules BM^D．Cai and Chen have proved this result in the case B=D=A．Secondly they have proved that if A has a nonzero left integral then A#A^*rat is a dense subring of Endk(A)．We prove that #(A,A) is a dense subring of Endk(Q),where Q is a certain subspace of #(A．A)under the condition that the antipode is bijective(see Theorem18).This condition is weaker than the condition that A has a nonzero integral.It is well known the antipode is bijective in case A has a nonzero integral．Furthermore if A has nonzero left integral,Q can be chosen to be A(see Corollary 19)and #(A,A)is both left and right primitive.Thus A#A^*rat #(A,A)-Endk(A)．Moreover we prove that the left singular ideal of the ring #(A,A)is zero．A corollary of this is a criterion for A with nonzero left integral to be finite-dimensional,namely the ring #(A,A)has a finite uniform dimension．     

ON fPP—Rings

in this psper,we investigate nore general rings than GPP-rings,called fPP-rings.First,we in-vestigate fPP-rings and their classical quotient quotient rings.We ptove (1) fPP-rings are f-quasi-regular rings.(2)R is a fPP-ring then Q(R) is fPP-ring.(3)R= iRi is a fPP-ring if and only if every Ri is a fPP-ring.Second,we present a characterization of fPP-ring via fP-injectivity,we prove that R is a fPP-ring if and only if every quotient module of a imjective R-module is fP-injectiv if and only ifevery quotient module of a P-injective R-module is fP-injective.Third,we study how fPP-rings are related to von Neu-mann regular rings,we prove that R is von Nevmann regular if and only if R is fPP-ring and for every α∈R,there is b∈E(R) and d∈R suth that α=f(α)b and f(α)=f^2(α) d for some f∈F(R).Finally,we give a example of fPP-ring which is not GPP-ring.     

On Flat and Coflat Modules

It is well known that a ring R is left hereditary iff every left ideal of R is projective, iff every submodule of a projective module is projective (ld R≤1), iff every quotient module of an injective module is injective (1cd R≤1), where 1d R and 1cd R means the left global dimension resp. codimension of the ring R. These rings may be generalized to those of weak left global dimension at most 1 (wld R≤1). The latter condition holds iff every left ideal of R is flat, iff every submodule     

Gorenstein flatness and injectivity over Gorenstein rings

Let R be a Gorenstein ring.We prove that if I is an ideal of R such that R/I is a semi-simple ring,then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical.In addition,we prove that if R→S is a homomorphism of rings and SE is an injective cogenerator for the category of left S-modules,then the Gorenstein flat dimension of S as a right R-module and the Gorenstein injective dimension of E as a left R-module are identical.We also give some applications of these results.     

半交换环的一个推广研究

In this paper, a generalization of the class of semicommutative rings is investigated.A ring R is called left GWZI if for any a ∈ R, l(a) is a GW-ideal of R. We prove that a ring R is left GWZI if and only if S3(R) is left GWZI if and only if Vn(R) is left GWZI for any n ≥ 2.     

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.     

The Auslander-type condition of triangular matrix rings

Let R be a left and right Noetherian ring and n,k be any non-negative integers.R is said to satisfy the Auslander-type condition G n (k) if the right flat dimension of the (i + 1)-th term in a minimal injective resolution of RR is at most i + k for any 0 i n 1.In this paper,we prove that R is Gn (k) if and only if so is a lower triangular matrix ring of any degree t over R.