On Regular Power-Substitution

The necessary and sufficient conditions under which a ring satisfies regular power-substitution are investigated. It is shown that a ring R satisfies regular powersubstitution if and only if a-b in R implies that there exist n ∈ N and a U E GLn （R） such that aU = Ub if and only if for any regular x ∈ R there exist m,n ∈ N and U ∈ GLn（R） such that x^mIn = xmUx^m, where a-b means that there exists x,y, z∈ R such that a =ybx, b = xaz and x= xyx = xzx. It is proved that every directly finite simple ring satisfies regular power-substitution. Some applications for stably free R-modules are also obtained.     

Some Characterizations of VNL Rings

A ring R is said to be von Newmann local (VNL) if for any a ∈ R, either a or 1 ?a is (von Neumann) regular. The class of VNL rings lies properly between exchange rings and (von Neumann) regular rings. We characterize abelian VNL rings. We also characterize and classify arbitrary VNL rings without an infinite set of orthogonal idempotents; and also the VNL rings having a primitive idempotent e such that eRe is not a division ring. We prove that a semiperfect ring R is VNL if and only if for any right uni-modular row (a ₁, a ₂) ∈ R ², one of the a _i 's is regular in R. Formal triangular matrix rings that are VNL are also characterized. As a corollary, it is shown that an upper triangular matrix ring T _n (R) is VNL if and only if n = 2 or 3 and R is a division ring.     

Study of Hopficity in Certain Classes of Rings

The main results proved in this paper are:

1. For any non-zero vector space V _Dover a division ring D, the ring R= End(V _D) is hopfian as a ring

2. Let Rbe a reduced π-regular ring &; B(R) the boolean ring of idempotents of R. If B(R) is hopfian so is R.The converse is not true even when Ris strongly regular.

3. Let Xbe a completely regular spaceC(X) (resp. C ^?(X)) the ring of real valued (resp. bounded real valued) continuous functions on X. Let Rbe any one of C(X) or C ^?(X). Then Ris an exchange ring if &; only if Xis zero dimensional in the sense of Katetov. for any infinite compact totally disconnected space X C(X) is an exchange ring which is not von Neumann regular.

4. Let Rbe a reduced commutative exchange ring. If Ris hopfian so is the polynomial ring R[T ₁,…,T _n] in ncommuting indeterminates over Rwhere nis any integer ≥ 1.

5. Let Rbe a reduced exchange ring. If Ris hopfian so is the polynomial ring R[T].     

On stable range conditions

In this paper, we show that a ring R satisfies unit 1-stable range if and only if a₁R + ? + a_mR = dR with m ≥ 2,a ₁, ?a_m ?R implies that there exist u₁ , ?u_m ? U(R) such that a₁u₁ +?+a_mu_m = d and an exchange ring R has stable range one if and only if a₁R+?+a_mR = dR with m ≥ 2,a ₁,?,a_m ? R implies that there exist unit-regular w ₁,?,w_m ? R such that a₁w₁ +?+ a_mw_m = d. Also we show that an exchange ring R satisfies the n-stable range condition if and only if a( ⁿR)+bR = dR with a ? Rⁿ,b,d ? R implies that there exist unimodular regular w ? ⁿ R and: y ? R such that aw+by = d.     

Generalized Stable Regular Rings

Abstract

In this paper we show that a regular ring R is a generalized stable ring if and only if for every x ∈ R, there exist a w ∈ K(R) and a group G in R such that wx ∈ G. Also we show that if R is a generalized stable regular ring, then for any A ∈ M _n (R), there exist right invertible matrices U ₁, U ₂ ∈ M _n (R) and left invertible matrices V ₁, V ₂ ∈ M _n (R) such that U ₁ V ₁ AU ₂ V ₂ = diag(e ₁,…, e _n ) for some idempotents e ₁,…, e _n  ∈ R.     

A reversal index for tournament matrices *

Given a tournament matrix T, its reversal indexi_R (T), is the minimum k such that the reversal of the orientation of k arcs in the directed graph associated with T results in a reducible matrix. We give a formula for i_R (T) in terms of the score vector of T which generalizes a simple criterion for a tournament matrix to be irreducible. We show that i_R (T)≤[(n?1)/2] for any tournament matrix T of order n, with equality holding if and only if T is regular or almost regular, according as n is odd or even. We construct, for each k between 1 and [(n?1)/2], a tournament matrix of order n whose reversal index is k. Finally, we suggest a few problems.     

When Are Covers Relative to Two Classes Isomorphic

Let ?₁ ? ?₂ be classes of right R-modules, we introduce ?₁-covering modules and projective modules relative to ?₂ to characterize the relations between the existences of ?₁-covers, ?₂-covers and the classes ?₁, ?₂. As corollaries, every module in ? _n is an ?-covering module if and only if every flat cover is an ? _n -cover for each right module if and only if R is a von Neumann regular ring whenever wD(R) < ∞; every flat right R-module is projective relative to 𝒫 if and only if every flat cover is a projective cover for each right module if and only if R is right perfect.     

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 the quotient between length and multiplicity

In this paper, the exchange ring R with the (general) ?₀-comparability is studied. A ring R is said to satisfy the general ?₀-comparability, if for any idempotent elements f, g ∈ R, there exist a positive integer n and a central idempotent element e ∈ R such that f Re ?_⊕ n[gRe] and gR(1 ? e) ?_⊕ n[f R(1 ? e)]. It is proved that the (general) ?₀-comparability for exchange rings is preserved under taking factor rings, matrix rings and corners. The ?₀-comparability condition for exchange rings R is characterized by the order structure of several partially ordered sets of ideals of R. For any exchange ring R with general ?₀-comparability and any proper ideal I of R not contained in J(R), it is proved that if I contains no nonzero central idempotents of R, then: 1) There exists an infinite set of nonzero idempotent elements {f _i ∣ i = 1,2, …} in I such that f ₁ R ?_⊕ f ₂ R ?_⊕ …, and n(f _n R) ?_⊕ R _R for all n ≥ 1; 2) For any m ≥ 1, there exist nonzero orthogonal idempotents e ₁, e ₂ …, e _m in I such that e ₁ R ⊕ e ₂ R ⊕ … ⊕ e _m R ?_⊕ I _R and e _i R ? e _j R for all i, j. For any exchange ring R with primitive factor rings artinian, if R satisfies the general ?₀-comparability, then in every ideal I of R not contained in J(R), there is a central idempotent element of R.     

The homomorphismM*⊗N → Hom (M,N)

In this note we prove two theorems. In theorem 1 we prove that if M andN are two non-zero reflexive modules of finite projective dimensions over a Gorenstein local ring, such that Hom (M, N) is a third module of syzygies, then the natural homomorphismM* ⊗N → Hom (M, N) is an isomorphism. This extends the result in [7]. In theorem 2, we prove that projective dimension of a moduleM over a regular local ringR is less than or equal ton if and only if Ext_R ⁿ (M, R) ⊗M → Ext_R ⁿ (M, M) is surjective; in which case it is actually bijective. This extends the usual criterion for the projectivity of a module.     

Exchange Rings Generated by Their Units

Let R be an exchange ring with primitive factors artinian. We prove that there exists a u∈U（R） such that 1R ± u ∈ U（R）, if and only if for any a ∈ R, there exists a u ∈ U（R） such that a ± u∈ U（R）. Phrthermore, we prove that, for any A ∈ Mn（R）（n ≥ 2）, there exists a U ∈ GLn（R） such that A ± U ∈ GLn（R）.     

Coherence and Generalized Morphic Property of Triangular Matrix Rings

Let R be a ring. R is left coherent if each of its finitely generated left ideals is finitely presented. R is called left generalized morphic if for every element a in R, l(a) = Rb for some b ∈ R, where l(a) denotes the left annihilator of a in R. The main aim of this article is to investigate the coherence and the generalized morphic property of the upper triangular matrix ring T _n (R) (n ≥ 1). It is shown that R is left coherent if and only if T _n (R) is left coherent for each n ≥ 1 if and only if T _n (R) is left coherent for some n ≥ 1. And an equivalent condition is obtained for T _n (R) to be left generalized morphic. Moreover, it is proved that R is left coherent and left Bézout if and only if T _n (R) is left generalized morphic for each n ≥ 1.     

EXCHANGE MORITA RINGS

In this paper we characterize the largest exchange ideal of a ring R as the set of those elements x ∈ R such that the local ring of R at x is an exchange ring. We use this result to prove that if R and S are two rings for which there is a quasi-acceptable Morita context, then R is an exchange ring if and only if S is an exchange ring, extending an analogue result given previously by Ara and the second and third authors for idempotent rings. We introduce the notion of exchange associative pair and obtain some results connecting the exchange property and the possibility of lifting idempotents modulo left ideals. In particular we obtain that in any exchange ring, orthogonal von Neumann regular elements can be lifted modulo any one-sided ideal.     

Diagonal reductions of matrices over exchange ideals

In this paper, we introduce related comparability for exchange ideals. Let I be an exchange ideal of a ring R. If I satisfies related comparability, then for any regular matrix A ∈ M _n (I), there exist left invertible U ₁; U ₂ ∈ M _n (R) and right invertible V ₁, V ₂ ∈ M _n (R) such that U ₁ V ₁ AU ₂ V ₂ = diag(e ₁,..., e _n ) for idempotents e ₁,..., e _n ∈ I.     

Closed Polynomials in Polynomial Rings over Unique Factorization Domains

Let R be a UFD, and let M(R, n) be the set of all subalgebras of the form R[f], where f ∈ R[x ₁,…, x _n ]?R. For a polynomial f ∈ R[x ₁,…, x _n ]?R, we prove that R[f] is a maximal element of M(R, n) if and only if it is integrally closed in R[x ₁,…, x _n ] and Q(R)[f] ∩ R[x ₁,…, x _n ] = R[f]. Moreover, we prove that, in the case where the characteristic of R equals zero, R[f] is a maximal element of M(R, n) if and only if there exists an R-derivation on R[x ₁,…, x _n ] whose kernel equals R[f].     

On the Multiplicative Monoid of n×n Matrices Over Artinian Chain Rings

Let R be an Artinian chain ring with a principal maximal ideal. We investigate properties of matrices over R and give matrix representations of R-submodules of R ⁿ first, then consider Green's relations, Green's relation equivalent classes, Schützenberger groups of 𝒟-classes, principal factors, and group ?-classes of the multiplicative monoid M _n (R) of n × n matrices over R. Furthermore, we show that M _n (R) is an eventually regular semigroup and derive basic numerical information of M _n (R) when R is finite.     

Onn-th commutators with nilpotent or regular values in rings

Then-th commutator for a,b in a ringR is defined inductively as follows: [a,b]₁=[a,b]=ab−ba and[a,b] _n=[[a,b]₋₁,b]. We characterize the ringsR without non-zero nil right ideals in which[a,b] _nis nilpotent or regular for alla,b∈R. We also examine the case whereR is a semiprime ring with involution in which[t ₁, t₂]_nis nilpotent or regular for all tracest ₁,t₂∈R.     

A Generalization of Divided Domains and Its Connection to Weak Baer Going-Down Rings

Many results on going-down domains and divided domains are generalized to the context of rings with von Neumann regular total quotient rings. A (commutative unital) ring R is called regular divided if each P ∈ Spec(R)?(Max(R) ∩ Min(R)) is comparable with each principal regular ideal of R. Among rings having von Neumann regular total quotient rings, the regular divided rings are the pullbacks K× _K/P D where K is von Neumann regular, P ∈ Spec(K) and D is a divided domain. Any regular divided ring (for instance, regular comparable ring) with a von Neumann regular total quotient ring is a weak Baer going-down ring. If R is a weak Baer going-down ring and T is an extension ring with a von Neumann regular total quotient ring such that no regular element of R becomes a zero-divisor in T, then R ? T satisfies going-down. If R is a weak Baer ring and P ∈ Spec(R), then R + PR _(P) is a going-down ring if and only if R/P and R _P are going-down rings. The weak Baer going-down rings R such that Spec(R)?Min(R) has a unique maximal element are characterized in terms of the existence of suitable regular divided overrings.