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AbstractWe study rings whose modules and module homomorphisms display behavior similar to that of sets and their maps. For example, whenever there is an epimorphism A→B, there is a monomorphism B→A (Artinian principal ideal rings (PIR) satisfy this property and its dual for all modules). As a byproduct of this framework, we prove that a ring every factor ring of which cogenerates its cyclic right modules (one-sided version of Kaplansky’s dual rings) is right Artinian and right serial. Consequently, R is an Artinian PIR if and only if every factor ring of R cogenerates its finitely generated right modules. These results can be viewed as partial answers to the CF problem, the FGF problem due to Faith and a question of Faith and Menal on strongly Johns rings. Some known results and the above one yield the following: A ring R is a direct sum of right Artinian right chain rings and Artinian PIR’s if and only if every factor ring of R cogenerates its (uniform) cyclic right modules (with nonzero socle); so, such rings coincide with the right CES-rings of Jain and Lopez-Pérmouth, rings whose factors are right CF and rings that satisfy the above mentioned property for their cyclic right modules A and B. Finally, a ring is either simple Artinian or a right Artinian right chain ring if and only if one of any two cyclic right modules embeds in the other. [ABSTRACT FROM AUTHOR]

Recall that a ring R is a Hilbert ring if any maximal ideal of R [ X ] contracts to a maximal ideal of R. The main purpose of this paper is to characterize the prime ideals of a commutative ring R which are traces of the maximal ideals of the polynomial ring R [ X ]. In this context, we prove that if p is a prime ideal of R such that R / p is a semi-local domain of (Krull) dimension ≤ 1 , then p is the trace of a maximal ideal of R [ X ]. Whereas, if R is Noetherian and either (dim (R / p) ≥ 2) or (the quotient field of R / p is algebraically closed, dim (R / p) = 1 and R / p is not semi-local), then p is never the trace of a maximal ideal of R [ X ]. Putting these results into use in investigating the Ext-index of Noetherian rings, we establish connections between the finiteness of the Ext-index of localizations of the polynomial rings R [ X ] and the finiteness of the Ext-index of localizations of R. This allows us to provide a new class of rings satisfying some known conjectures on Ext-index of Noetherian rings as well as to build bridges between these conjectures. [ABSTRACT FROM AUTHOR]

In the first part of the paper, we provide a fairly complete description of a simple ring R with involution f such that relative to f the traces of all elements of R form a commutative subset of R. This description is based on the characteristic of R. While the case of characteristic not 2 readily follows from current results in the literature, by contrast, the opposite case of characteristic 2 requires markedly more work. In the rest of the paper, the results carried out earlier are put to work to delineate the structure of a simple Artinian ring R with involution f, which is equipped with a given non central multiplicative subset M stabilized by f and preserved by all inner automorphisms of R, and such that the traces relative to f of all elements of M form a commutative subset of R. [ABSTRACT FROM AUTHOR]

It is proved that a ring A is a right or left Noetherian, right distributive, centrally essential ring if and only if A = A 1 × ⋯ × A n , where each of the rings A i is either a commutative Dedekind domain or a left and right Artinian, left and right uniserial ring. [ABSTRACT FROM AUTHOR]

In this paper, we relate some properties of non-comaximal graph of ideals of a commutative ring with identity with the properties of the ring. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and let (x) be the principal ideal generated by x ∈ R. Let Ω (R) ∗ be the set of all nontrivial principal ideals of R. The reduced cozero-divisor graph Γ r (R) of R is an undirected simple graph with Ω (R) ∗ as the vertex set and two distinct vertices (x) and (y) in Ω (R) ∗ are adjacent if and only if (x) ⊈ (y) and (y) ⊈ (x) . In this paper, we characterize all classes of commutative Artinian non-local rings for which the reduced cozero-divisor graph has genus at most one. [ABSTRACT FROM AUTHOR]

A ring R is said to have right restricted minimum condition ( r. RMC , for short) if R/A is an Artinian right R-module for any essential right ideal A of R. We study the RMC for matrix extensions of a ring R and for the group-ring RG. Among other things, it is proven that having r.RMC is a Morita invariant property for rings. For n ≥ 2 , the upper triangular n by n matrix ring over R has r.RMC if and only if RR is Artinian. If G is a not torsion abelian group then RG has r.RMC if and only if R is a semisimple ring and G ≃ H ⊕ Z where H is a finite group whose order is invertible in R. If X is a completely regular topological space, then the ring C(X) has RMC if and only if X is finite. Many examples of rings with r.RMC are presented. [ABSTRACT FROM AUTHOR]

In this paper; we will study the relationship between simple submodules of M and the C-ring. A ring R is called C-ring if every torsion module M over R has simple submodules. We investigated that if M is a torsion module over a semisimple ring; so, R is C-ring. Also, we study C-ring when R is Noetherian and T(M) = M such that any direct sum of modules with (SIP) has (SSP) implies that R is C- ring. We proved that if M is a torsion module and M = W + W′, this means R is C-ring where W and W′ are direct sum of M. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with non-zero identity. The annihilator-inclusion ideal graph of R, denoted by R, is a graph whose vertex set is the of all non-zero proper ideals of R and two distinct vertices I and J are adjacent if and only if either Ann(I) J or Ann(J) I. The purpose of this paper is to provide some basic properties of the graph R. In particular, shows that R is a connected graph with diameter at most three, and has girth 3 or 1. Furthermore, is determined all isomorphic classes of non-local Artinian rings whose annihilator-inclusion ideal graphs have genus zero or one. [ABSTRACT FROM AUTHOR]

We introduce Rad-discrete and quasi-Rad-discrete modules as a proper generalization of (quasi) discrete modules, and provide various properties of these modules. We prove that a direct summand of a (quasi) Rad-discrete module is (quasi) Rad-discrete. We show that every projective R-module is (quasi) Rad-discrete if and only if R is left perfect. We also prove that, over a commutative Noetherian ring R, every quasi-Rad-discrete R-module is the direct sum of local R-modules if and only if R is Artinian. Finally, we investigate self-projective Rad-discrete modules and π -projective quasi-Rad-discrete modules over Dedekind domains. [ABSTRACT FROM AUTHOR]

Let X be an arbitrary topological space. F(X) denotes the set of all real-valued functions on X and C(X)F denotes the set of all f ∈ F(X) such that f is discontinuous at most on a finite set. It is proved that if r is a positive real number, then for any f ∈ C(X)F which is not a unit of C(X)F there exists g ∈ C(X)F such that g ≠ 1 and f = gr f. We show that every member of C(X)F is continuous on a dense open subset of X if and only if every non-isolated point of X is nowhere dense. It is shown that C(X)F is an Artinian ring if and only if the space X is finite. We also provide examples to illustrate the results presented herein. [ABSTRACT FROM AUTHOR]

We realize Leavitt ultragraph path algebras as partial skew group rings. Using this realization we characterize artinian ultragraph path algebras and give simplicity criteria for these algebras. [ABSTRACT FROM AUTHOR]

Let (A, M) be any finite local (nonzero commutative unital) ring. Then the set of A-algebra isomorphism classes represented by (equivalently, consisting of) rings B such that A ⊂ B is a decomposed (minimal ring) extension is in one-to-one correspondence with the collection of sets { I 1 , I 2 } of ideals I 1 and I 2 of A such that M = I 1 ⊕ I 2 (internal direct sum), with any such { I 1 , I 2 } corresponding to the class represented by the ring B = A / I 1 × A / I 2. If A is not a field, then the (finite) set of A-algebra isomorphism classes represented by (equivalently, consisting of) rings B such that A ⊂ B is a ramified (minimal ring) extension has cardinality at least 2; known examples show that this bound cannot be improved. If X denotes an indeterminate over the field F 4 , then R : = F 2 + X F 4 [ X ] / (X 2) is of minimal cardinality in the class of finite local rings A such that A is not a field and the A-algebra isomorphism classes represented by a ramified extension of A constitute fewer than half of the (finitely many) A-algebra isomorphism classes represented by a minimal ring extension of A. [ABSTRACT FROM AUTHOR]

Let R be a commutative unital ring, and let Spec (R) : = P (R) be the collection of prime ideals of R. Further, let N P (R) denote the collection of proper nonprime ideals of R and set N P (R) 1 : = N P (R) ∪ { R }. In this note, we investigate some closure properties relative to these collections of ideals with respect to ideal addition and multiplication. Among the main objects of our study are rings R for which P (R) + N P (R) ⊆ P (R) , N P (R) 1 + N P (R) 1 ⊆ N P (R) 1 , or P (R) · P (R) ⊆ P (R). Directions for further research are indicated. Communicated by Sudarshan Kumar Sehgal [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with nonzero identity and S ⊆ R be a multiplicatively closed subset. In this paper, we study S -Artinian rings and finitely S -cogenerated rings. A commutative ring R is said to be an S -Artinian ring if for each descending chain of ideals { I n } n ∈ ℕ of R , there exist s ∈ S and k ∈ ℕ such that s I k ⊆ I n for all n ≥ k. Also, R is called a finitely S -cogenerated ring if for each family of ideals { I α } α ∈ Δ of R , where Δ is an index set, ⋂ α ∈ Δ I α = 0 implies s (⋂ α ∈ Δ ′ I α) = 0 for some s ∈ S and a finite subset Δ ′ ⊆ Δ. Moreover, we characterize some special rings such as Artinian rings and finitely cogenerated rings. Also, we extend many properties of Artinian rings and finitely cogenerated rings to S -Artinian rings and finitely S -cogenerated rings. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian domain, M a nonzero R -module of finite injective dimension, and I be a nonzero ideal of R. In this paper, we prove that whenever injdim M = c d (I , M) = t , then the annihilator of H I t (M) is zero. Also, we calculate the annihilator of H I n + t (N , M) for finitely generated R -modules N and M with conditions projdim N = n < ∞ and injdim M = t < ∞. Moreover, if (R , 𝔪) is a regular Noetherian local ring and 𝔭 ∈ Spec (R) such that d = dim R 𝔭 ≥ 2 , then we show that there exists an ideal J of R such that 𝔭 ⊆ J , h t J 𝔭 = 1 and J n H 𝔪 1 (R 𝔭 n ) = 0. [ABSTRACT FROM AUTHOR]

It is proved that a ring R is a right uniserial, right Noetherian centrally essential ring if and only if R is a commutative discrete valuation domain or a left and right Artinian, left and right uniserial ring. It is also proved that there exist non-commutative uniserial Artinian centrally essential rings. [ABSTRACT FROM AUTHOR]

The possibility to generalize the notion of a linear recurrent sequence (LRS) over a commutative ring to the case of a LRS over a non-commutative ring is discussed. In this context, an arbitrary bimodule AMB over left- and right-Artinian rings A and B, respectively, is associated with the equivalent bimodule of translations CMZ, where C is the multiplicative ring of the bimodule AMB and Z is its center, and the relation between the quasi-Frobenius conditions for the bimodules AMB and CMZ is studied. It is demonstrated that, in the general case, the fact that AMB is a quasi-Frobenius bimodule does not imply the validity of the quasi-Frobenius condition for the bimodule CMZ. However, under some additional assumptions it can be shown that if CMZ is a quasi-Frobenius bimodule, then the bimodule AMB is quasi-Frobenius as well. [ABSTRACT FROM AUTHOR]

Abstract We show that, in the context of Reverse Mathematics, WK L 0 (Weak König's Lemma) implies the statement AR T 0 which says that every Artinian ring is Noetherian, over RC A 0 (Recursive Comprehension Axiom). To achieve this goal, we prove a general Computable Full Structure Theorem for computable Artinian rings similar to the classical version found in most Algebra texts. [ABSTRACT FROM AUTHOR]

This paper introduces the notion of dual of automorphism extendable modules. A module M is called automorphism-extendable if for every submodule N of M, every automorphism of N can be extended to an endomorphism of M. We call a module M a dual automorphism-extendable module if whenever K is a submodule of M, then every automorphism ν:M∕K → M∕K lifts to an endomorphism 휃 of M. In this paper we give various examples of dual automorphism-extendable modules and study their properties. In particular, we prove that every dual automorphism-extendable module is a D3-module. It is shown that over a right artinian ring R, an R-module with hollow modules Mi is dual automorphism-extendable if and only if M is quasi-projective. [ABSTRACT FROM AUTHOR]

Let α = { α g : R g − 1 → R g } g ∈ mor ( G ) be a partial action of a groupoid G on a (not necessarily associative) ring R and let S = R ⋆ α G be the associated partial skew groupoid ring. We show that if α is global and unital, then S is left (right) artinian if and only if R is left (right) artinian and R g = { 0 } , for all but finitely many g ∈ mor ( G ) . We use this result to prove that if α is unital and R is alternative, then S is left (right) artinian if and only if R is left (right) artinian and R g = { 0 } , for all but finitely many g ∈ mor ( G ) . This result applies to partial skew group rings, in particular. Both of the above results generalize a theorem by J. K. Park for classical skew group rings, i.e. the case when R is unital and associative, and G is a group which acts globally on R . We provide two additional applications of our main results. Firstly, we generalize I. G. Connell's classical result for group rings by giving a characterization of artinian (not necessarily associative) groupoid rings. This result is in turn applied to partial group algebras. Secondly, we give a characterization of artinian Leavitt path algebras. At the end of the article, we relate noetherian and artinian properties of partial skew groupoid rings to those of global skew groupoid rings, as well as establish two Maschke-type results, thereby generalizing results by M. Ferrero and J. Lazzarin for partial skew group rings to the case of partial skew groupoid rings. [ABSTRACT FROM AUTHOR]

The regular graph of ideals of the commutative ring R, denoted by Γreg(R), is a graph whose vertex set is the set of all non-trivial ideals of R and two distinct vertices I and J are adjacent if and only if either I contains a J-regular element or J contains an I-regular element. In this paper, it is proved that the radius of Γreg(R) equals 3. The central vertices of Γreg(R) are determined, too. [ABSTRACT FROM AUTHOR]

We investigate properties of FC-rings (i.e. rings R in which the centralizer CR (a) of any element a ∊ R is of finite index in R)and, in particular, characterize left Artinian rings with a finite set of all derivations Der R (respectively inner derivations Der R (respectively inner derivations IDer R).We show that if R is a Jacobson radical ring in which its adjoint group R° has a finite number of conjugacy classes, then R=Rp1 ⊕....⊕ Rpt ⊕ D is a ring direct sum of Jacobson radical rings Rpi and D, where the additive group D+ is a torsion-free divisible group, the adjoint group D° is a group with a finite number of conjugacy classes, R+pi is a finite pi- group (i=l,....,t) and p1,....,pt are pairwise distinct primes. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Γ'(R) is a graph with vertex set W*(R), where W*(R) is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b are adjacent if and only if a ∉ (b) and b ∉ (a). In this paper, we study the primitive cycle of cozero-divisor graphs and we determine all Artinian rings with claw-free or triangle-free cozero-divisor graphs. Also, we investigate all Artinian rings whose cozero-divisor graphs are C4-free. [ABSTRACT FROM AUTHOR]

For certain classes 𝒞 ofR-modules, including singular modules or modules with locally Krull dimensions, it is investigated when every module in 𝒞 with a finitely generated essential submodule is finitely generated. In case 𝒞 = Mod-R, this means E(M)/Mis Noetherian for any finitely generated moduleMR. RingsRwith latter property are studied and shown that they form a class 𝒬 properly between the class of pure semisimple rings and the class of certain max rings. Duo rings in 𝒬 are precisely Artinian rings. IfRis a quasi continuous ring in 𝒬 thenR ≃ A ⊕ TwhereAis a semisimple Artinian ring andT ∈ 𝒬 with Z(TT) ≤essTT. [ABSTRACT FROM AUTHOR]

We study the conjecture that, ifI,Jare μ −primary in a regular local ring (R, μ) with dim(R) = n, thenneeds at leastngenerators, and a related conjecture about the number of generators of the first Koszul homology module of an Artinian local ring (A,m). In this manuscript, we focus our attention on the complete intersection defect of the Artinian ring and its quotient by the Koszul elements. We prove that the number of generators of the first Koszul homology module ofx1,…,xn ∈ mon an Artinian local ring (A,m) is at least, where cidAdenotes the complete intersection defect of the Artinian local ringA. [ABSTRACT FROM AUTHOR]

In this paper, we define coprimely structured rings, which are the generalization of strongly 0-dimensional rings. Furthermore, we investigate coprimely structured rings and give some relations between other rings such as Artinian rings, strongly 0-dimensional rings, and h-local domains. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and $$\mathrm{Nil}(R)$$ be the set of nilpotent elements of R. The nil-graph of ideals of R is defined as the graph $${\mathbb {AG}}_N(R)$$ whose vertex set is $$\{I:\ (0)\ne I\lhd R$$ , and there exists a nontrivial ideal J such that $$IJ\subseteq \mathrm{Nil}(R)\}$$ and two distinct vertices I and J are adjacent if and only if $$IJ\subseteq \mathrm{Nil}(R)$$ . Here, some graph properties of $${\mathbb {AG}}_N(R)$$ are studied. For instance, some bounds for the diameter, girth, and radius of $${\mathbb {AG}}_N(R)$$ are given. In case that $${\mathbb {AG}}_N(R)$$ is a finite graph, it is proved that the center and median of $${\mathbb {AG}}_N(R)$$ coincide. Furthermore, we determine when the edge chromatic number of $${\mathbb {AG}}_N(R)$$ equals its maximum degree. Also, for every ring R, it is shown that both the clique number and vertex chromatic number of $${\mathbb {AG}}_N(R)$$ equal $$n+t$$ , where n is the number of minimal prime ideals of R and t is the number of nonzero ideals of R which are contained in $$\mathrm{Nil}(R)$$ . [ABSTRACT FROM AUTHOR]

Given a commutative ring with identity and an -module , a subset of is a cyclic covering of , if this module is the union of the cyclic submodules , where . Such covering is said to be irredundant, if no proper subset of is a cyclic covering of . In this work, an irredundant cyclic covering of is constructed for every Artinian commutative ring . As a consequence, a cyclic covering of minimal cardinality of is obtained for every finite commutative ring , extending previous results in the literature. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring. In this paper, we develop the existence of direct limits of finite products of fields as subrings of R. [ABSTRACT FROM AUTHOR]

The zero-divisor graph of a commutative ring R with respect to nilpotent elements is a simple undirected graph Γ*N(R) with vertex set ZN(R)*, and two vertices x and y are adjacent if and only if xy is nilpotent and xy ≠ 0, where ZN(R) = {x ∈ R : xy is nilpotent, for some y ∈ R*}. In this paper, we investigate the basic properties of Γ*N(R). We discuss when it will be Eulerian and Hamiltonian. We further determine the genus of Γ*N(R). [ABSTRACT FROM AUTHOR]

Suppose F is a totally ordered field equipped with its order topology and X a completely F-regular topological space. Suppose Ƥ is an ideal of closed sets in X and X is locally-Ƥ. Let CƤ(X, F) = {ƒ : X → F | ƒ is continuous on X and its support belongs to Ƥ} and C∞Ƥ (X, F) = .... Then CƤ(X, F) is a Noetherian ring if and only if C∞Ƥ (X, F) is a Noetherian ring if and only if X is a finite set. The fact that a locally compact Hausdorff space X is finite if and only if the ring CK(X) is Noetherian if and only if the ring C∞(X) is Noetherian, follows as a particular case on choosing F = R and Ƥ = the ideal of all compact sets in X. On the other hand if one takes F = R and Ƥ = the ideal of closed relatively pseudocompact subsets of X, then it follows that a locally pseudocompact space X is finite if and only if the ring Cψ(X) of all real valued continuous functions on X with pseudocompact support is Noetherian if and only if the ring ... is pseudocompact} is Noetherian. Finally on choosing F = R and Ƥ = the ideal of all closed sets in X, it follows that: X is finite if and only if the ring C(X) is Noetherian if and only if the ring C*(X) is Noetherian. [ABSTRACT FROM AUTHOR]

Let $${\mathcal{P}}$$ be an ideal of closed quotients of a completely regular frame L and $${\mathcal{R}_{\mathcal{P}}(L)}$$ the collection of all functions in the ring $${\mathcal{R}(L)}$$ whose support belong to $${\mathcal{P}}$$ . We show that $${\mathcal{R}(L)}$$ is a Noetherian ring if and only if $${\mathcal{R}(L)}$$ is an Artinian ring if and only if L is a finite frame. Using this result, we next show that if $${\mathcal{P}}$$ is the ideal of all compact closed quotients of L and L is $${\mathcal{P}}$$ -continuous, then $${\mathcal{R}_{\mathcal{P}}(L)}$$ is a Noetherian ring if and only if L is finite. Moreover, we show that L is a P-frame if and only if each ideal of $${\mathcal{R}(L)}$$ is of the form $${\mathcal{R}_{\mathcal{P}}(L)}$$ for some choice of $${\mathcal{P}}$$ . We furnish equivalent conditions for $${\mathcal{R}_{\mathcal{P}}(L)}$$ to be a prime ideal, a free ideal, and an essential ideal of $${\mathcal{R}(L)}$$ separately in terms of the cozero elements of L. Finally, we show that L is basically disconnected if and only if $${\mathcal{R}(L)}$$ is a coherent ring. [ABSTRACT FROM AUTHOR]

LetVbe a finite-dimensional vector space over a fieldk, and letWbe a 1-dimensionalk-vector space. Let ⟨,⟩:V × V → Wbe a symmetric bilinear form. Then ⟨,⟩ is calledanisotropicif for all nonzerov ∈ Vwe have ⟨ v,v ⟩ ≠ 0. Motivated by a problem in algebraic number theory, we give a generalization of the concept ofanisotropyto symmetric bilinear forms on finitely generated modules over artinian principal ideal rings. We will give many equivalent definitions of this concept of anisotropy. One of the definitions shows that a form is anisotropic if and only if certain forms on vector spaces are anisotropic. We will also discuss the concept ofquasi-anisotropyof a symmetric bilinear form, which has no vector space analogue. Finally, we will discuss theradical rootof a symmetric bilinear form, which does not have a vector space analogue either. All three concepts have applications in algebraic number theory. [ABSTRACT FROM AUTHOR]

For a space, we investigate its CJL (cohomology jump loci), sitting inside varieties of representations of the fundamental group. To do this, for a CDG (commutative differential graded) algebra, we define its CJL, sitting inside varieties of flat connections. The analytic germs at the origin 1 of representation varieties are shown to be determined by the Sullivan 1-minimal model of the space. Up to a degree q, the two types of CJL have the same analytic germs at the origins, when the space and the algebra have the same q-minimal model. We apply this general approach to formal spaces (obtaining the degeneration of the Farber-Novikov spectral sequence), quasi-projective manifolds, and finitely generated nilpotent groups. When the CDG algebra has positive weights, we elucidate some of the structure of (rank one complex) topological and algebraic CJL: all their irreducible components passing through the origin are connected affine subtori, respectively rational linear subspaces. Furthermore, the global exponential map sends all algebraic CJL into their topological counterpart. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with identity, and let I be an (left, right, two-sided) ideal of R. Say that I is small if |I| < |R| and large if |R/I| < |R|. In this paper, we present results on small and large ideals. In particular, we study their interdependence and how they influence the structure of R. Conversely, we investigate how the ideal structure of R determines the existence of small and large ideals. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and A(R) be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R)* = A(R)\{(O)} and two distinct vertices I and J are adjacent if and only if I J = (0). Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and ω(AG(R)) = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite. [ABSTRACT FROM AUTHOR]

Abstract: Let R be a semiprimitive algebra, d its algebraic derivation and the subalgebra of constants of d. It is proved that the Jacobson radical of is nilpotent. It is also shown that the following properties are equivalent: is semilocal; R is semisimple Artinian; is left and right Artinian. [Copyright &y& Elsevier]

Let R be a ring, R is called a right ACS ring if the right annihilator of each element is essential in a direct summand of R. We prove that right CF ring is right artinian under right ACS condition. In particular, FGF conjecture is true under right ACS condition. We also give some conditions under which right noetherian left P-injective and left ACS ring is QF. [ABSTRACT FROM AUTHOR]

Let R be a ring with non-zero identity and U( R) be the group of units of R. The unit graph of R, denoted by G( R), is a graph defined on the elements of R, and two distinct vertices r and s are adjacent if and only if r+ s∈ U( R). We investigate connectivity, diameter and the girth of the unit graph of a left Artinian ring. Also, by providing an algorithm, we determine when the unit graph of a finite ring is Hamiltonian. [ABSTRACT FROM AUTHOR]

Let S be a finitely generated standard multigraded algebra over an Artinian local ring A, and let M be a finitely generated multigraded S-module. This paper answers the question when mixed multiplicities of M are positive and characterizes them in terms of lengths of A-modules. As an application, we get interesting results on mixed multiplicities of ideals, and recover some early results. [ABSTRACT FROM AUTHOR]

A proper subring S of a ring R is said to be maximal if there is no subring of R properly between S and R. If R is a noetherian domain with |R| > 2ℵ0, then |(R)| ≤ |(R)|, where (R) is the set of maximal subrings of R. A useful criterion for the existence of maximal subrings in any ring R is also given. It is observed that if S is a maximal subring of a ring R, then S is artinian if and only if R is artinian and integral over S. Surprisingly, it is shown that any infinite direct product of rings has always maximal subrings. Finally, maximal subrings of zero-dimensional rings are also investigated. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and Max ( R) be the set of maximal ideals of R. The regular digraph of ideals of R, denoted by $\overrightarrow{\Gamma_{\mathrm{reg}}}(R)$ , is a digraph whose vertex set is the set of all non-trivial ideals of R and for every two distinct vertices I and J, there is an arc from I to J whenever I contains a J-regular element. The undirected regular (simple) graph of ideals of R, denoted by Γ( R), has an edge joining I and J whenever either I contains a J-regular element or J contains an I-regular element. Here, for every Artinian ring R, we prove that |Max ( R)|−1≦ ω(Γ( R))≦|Max ( R)| and $\chi(\Gamma_{\mathrm{ reg}}(R)) = 2|\mathrm{Max}\, (R)| -k-1$ , where k is the number of fields, appeared in the decomposition of R to local rings. Among other results, we prove that $\overrightarrow{\Gamma_{\mathrm{ reg}}}(R)$ is strongly connected if and only if R is an integral domain. Finally, the diameter and the girth of the regular graph of ideals of Artinian rings are determined. [ABSTRACT FROM AUTHOR]

Given a significative class ℱ of commutative rings, we study the precise conditions under which a commutative ring R has an ℱ-envelope. A full answer is obtained when ℱ is the class of fields, semisimple commutative rings or integral domains. When ℱ is the class of Noetherian rings, we give a full answer when the Krull dimension of R is zero and when the envelope is required to be epimorphic. The general problem is reduced to identifying the class of non-Noetherian rings having a monomorphic Noetherian envelope, which we conjecture is the empty class. [ABSTRACT FROM AUTHOR]

Given a ring R, we study the bimodules M for which the trivial extension R ∝ M is morphic. We obtain a complete characterization in the case where R is left perfect, and we prove that R ∝ Q/R is morphic when R is a commutative reduced ring with classical ring of quotients Q. We also extend some known results concerning the connection between morphic rings and unit regular rings. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the intersection graph of nontrivial left ideals of a ring . We characterize the rings for which the graph is connected and obtain several necessary and sufficient conditions on a ring such that is complete. For a commutative ring with identity, we show that is complete if and only if is also so. In particular, we determine the values of for which is connected, complete, bipartite, planar or has a cycle. Next, we characterize finite graphs which arise as the intersection graphs of and determine the set of all non-isomorphic graphs of for a given number of vertices. We also determine the values of for which the graph of is Eulerian and Hamiltonian. [Copyright &y& Elsevier]

An example is given of an Artinian local (commutative unital) ring R and a finite group G acting on R (via ring automorphisms) such that RG is not an Artinian ring; in this example, |G| necessarily fails to be a unit of R. Also, an example is constructed of a ring R on which an infinite cyclic group G acts such that the ring extension RG ⊆ R does not satisfy the going-down property GD; in this second example, the G-action on R is necessarily not locally finite, and it can be arranged that R is a monoid ring with any desired infinite cardinality and that RG is an integral domain with any (prime or 0) characteristic. [ABSTRACT FROM AUTHOR]

Abstract: For R an artinian ring and G a group, we characterize when RG is a principal ideal ring. In the case when G is finite (and R artinian), this yields a characterization of when RG is a left and right morphic ring. This extends work done by Passman, Sehgal and Fisher on principal ideal group rings when the coefficient ring is a field, and work of Chen, Li, and Zhou on morphic group rings. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to give necessary and sufficient conditions for rings for which every right self-small module is finitely generated. It is proved that: semi-simple rings, commutative perfect rings and right nonsingular Σ-extending rings have this property; a right nonsingular semi-prime ring has the property if and only if it is semi-simple; a commutative noetherian ring has the property if and only if it is artinian. [Copyright &y& Elsevier]

A module M over a ring R is called a lifting module if every submodule A of M contains a direct summand K of M such that A / K is a small submodule of M / K (e.g., local modules are lifting). It is known that a (finite) direct sum of lifting modules need not be lifting. We prove that R is right Noetherian and indecomposable injective right R -modules are hollow if and only if every injective right R -module is a direct sum of lifting modules. We also discuss the case when an infinite direct sum of finitely generated modules containing its radical as a small submodule is lifting. [ABSTRACT FROM AUTHOR]