Search

We study the primeness of noncommutative polynomials on prime rings. Let R be a prime ring with extended centroid C, ρ a right ideal of R, f(X1,…,Xt) a noncommutative polynomial over C, which is not a polynomial identity (PI) for ρ, and a,b ∈ R∖{0}. Then af(x1,…,xt)b = 0 for all x1,…,xt ∈ ρ if and only if one of the following holds: (i) aρ = 0; (ii) ρC = eRC for some idempotent e ∈ RC and b ∈ ρC such that either f(X1,…,Xt)Xt+1 is a PI for ρ or f(X1,…,Xt) is central-valued on eRCe and ab = 0. We then apply the result to higher commutators of right ideals. Some results of the paper are also studied from the view of point of the notion of X-primeness of rings. [ABSTRACT FROM AUTHOR]

This paper describes properties of three certain classes of modules M over a ring R determined by conditions on isomorphic direct summands ( ≲ ⊕ ): The condition that whenever (I m λ ≅) M / Ker λ ≲ ⊕ M then Ker λ and I m λ are direct summands of M for any endomorphism λ ∈ End (M) (kernel-endoregular modules). The condition that if M / A ≅ B where A, B ≲ ⊕ M then M / B ≅ A (iso-summand-morphic modules). The condition if M / A ≅ B where A, B ≤ ⊕ M , then M / B ≅ A (summand-morphic modules) which is precisely the internal cancellation property for modules. [ABSTRACT FROM AUTHOR]

An error in Corollary 9.32 of [C. Faith, Rings and Things and a Fine Array of Twentieth Century Associative Algebra, Mathematical Surveys and Monographs, Vol. 65 (American Mathematical Society, Providence, RI, 2004)], motivated us to consider again FPF rings which were initiated by Faith in the 1970s. In this paper, it is shown that a commutative ring R is reduced FPF if and only if it is Π-semihereditary. We show that when a semiperfect ring with a strongly right bounded basic ring with right and left Ore conditions, is an FPF ring. After some general results, the article focuses on rings of continuous functions. We give some algebraic characterizations for a C(X) to be FPF and retrieve a result of Jorge Martinez. Also, we show that a space X is fraction-dense if and only if Qcl(X) is a continuous ring. [ABSTRACT FROM AUTHOR]

A ring R is called left fusible if every nonzero element is the sum of a left zero-divisor and a non-left zero-divisor, and R is called uniquely left fusible if for any a ∈ R there exists a unique left zero-divisor z such that a – z is non-left zero-divisor. We show that a left fusible ring R is uniquely left fusible if and only if either R is a domain or R has a unique non-left zero-divisor element. [ABSTRACT FROM AUTHOR]

We characterize bilinear functionals ϕ on a symmetric algebra A satisfying the two-sided zero product property (the 2-zpp, i.e., ϕ (x , y) = 0 whenever x y = y x = 0). If A is also a zero product determined algebra and if every derivation of the algebra A is inner, then A is a 2-zpd algebra (i.e., every bilinear functional on A satisfying the 2-zpp is of the form (x , y) ↦ τ 1 (x y) + τ 2 (y x) for x , y ∈ A , where τ 1 , τ 2 are linear functionals on A). Conversely, if A is a finite-dimensional 2-zpd algebra, then the derivations of A are characterized, that is, given any derivation d of the algebra A , there exists a ∈ A such that, for all x ∈ A , d (x) − [ a , x ] lies in the Jacobson radical of A. Finally, we determine all bilinear functionals satisfying the 2-zpp on a specific zpd symmetric algebra and hence decide whether such an algebra is 2-zpd. [ABSTRACT FROM AUTHOR]

In this paper, we study the behavior of endomorphism rings of indecomposable (uniform) quasi-injective modules. A very natural question here is, for a morphism f : A → B , with A , B indecomposable (uniform) quasi-injective right R -modules, and g : E (A) → E (B) an extension of f where E (−) denotes the injective hull, what is the relation between kernels of f and g , their monogeny classes and their upper parts? [ABSTRACT FROM AUTHOR]

A right R -module M is called a U -module if, whenever A and B are submodules of M with A ≅ B and A ∩ B = 0 , there exist two direct summands K and T of M such that A ⊆ ess K , B ⊆ ess T and K ⊕ T ⊆ ⊕ M. The class of U -modules is a strict and simultaneous generalization of quasi-continuous, square-free and automorphism-invariant modules. In this paper we study the rings whose cyclics are U -modules, extending many of the known results on the subject and providing new ones. [ABSTRACT FROM AUTHOR]

The aim of the paper is to characterize the equation 1 + Δ (R) = U (R) , where J (R) ⊆ Δ (R) = { x ∈ R : x + u ∈ U (R)   for all   u ∈ U (R) } that is the largest Jacobson radical subring of R and U (R) is the set of invertible elements of a ring R. We show that this equation is closely related to U J -rings and rings whose elements can be written as the sum of an idempotent and an element from Δ (R). After presenting several characterizations and properties of this equation, we consider the rings satisfying the equation 1 + Δ (R) = U (R) within many well-studied classes of rings. Finally, we close the paper with group rings. [ABSTRACT FROM AUTHOR]

The object of this article is associate to automorphism-invariant modules that are invariant under any automorphism of their injective hulls with cyclic modules and cyclic modules have cyclic automorphism-invariant hulls. The study of the first sequence allows us to characterize rings whose cyclic right modules are automorphism-invariant and to show that if R is a right Köthe ring, then R is an Artinian principal left ideal ring in case every cyclic right R-module is automorphism-invariant. The study of the second sequence leads us to consider a generalization of hypercyclic rings that are each cyclic R-module has a cyclic automorphism-invariant hull. Such rings are called right a-hypercyclic rings. It is shown that every right a-hypercyclic ring with Krull dimension is right Artinian. [ABSTRACT FROM AUTHOR]

Recall that a ring satisfies the 2-sum property if each of its elements is a sum of two units. Here a ring R is said to satisfy the binary 2-sum property if, for any a, b in R, there exists a unit u of R such that both a - u and b - u are units. A well-known result, due to Goldsmith, Pabst and Scot, states that a semilocal ring satisfies the 2-sum property iff it has no image isomorphic to Z 2 . It is proved here that a semilocal ring satisfies the binary 2-sum property iff it has no image isomorphic to Z 2 or Z 3 or M 2 (Z 2) . [ABSTRACT FROM AUTHOR]

In analogy to the elementwise definitions of UU- and UJ-rings, a ring R is called UNJ if 1 + N (R) + J (R) = U (R). After presenting several characterizations and properties, we consider the UNJ-rings within many well-studied classes of rings. In particular, we examine Dedekind finite rings, 2-primal rings, (semi)regular rings, π -regular rings and rings that satisfy the identity x 2 = x. Finally, we conclude this paper with group UNJ-rings. [ABSTRACT FROM AUTHOR]

For n ≥ 2 and for a ring R, the notation Pn(R) means that an − a is nilpotent for all a ∈ R, and Qn(R) means that R/J(R) has identity xn = x and J(R) is nil, where J(R) is the Jacobson radical of R. In this paper, rings R for which Pn(R) holds are completely characterized for integers n up to 12, for n = 2k with k odd, and for n = 2kp, wherek > 0 and p = 3, 5, 7 or 9. For an arbitrary integer n ≥ 2, we prove that Pn(R) ⇔ Qn(R) for all rings R if and only if n is even with n ≡ 1 (mod 3), or n is odd with n ≢ 1 (mod 3) and n ≢ 1 (mod 8). [ABSTRACT FROM AUTHOR]

A ring R is called eversible if every left zero-divisor in R is also a right zero-divisor and conversely. This class of rings is a natural generalization of reversible rings. It is shown that every eversible ring is directly finite, and a von Neumann regular ring is directly finite if and only if it is eversible. We give several examples of some important classes of rings (such as local, abelian) that are not eversible. We prove that R is eversible if and only if its upper triangular matrix ring T n (R) is eversible, and if M n (R) is eversible then R is eversible. [ABSTRACT FROM AUTHOR]

Rings all of whose right ideals are automorphism-invariant are called right -rings. In the present paper, we study rings having the property that every right cyclic module is dual automorphism-invariant. Such rings are called right -rings. We obtain some of the relationships -rings and -rings. We also prove that; (i) A semiperfect ring is a right -ring if and only if any right ideal in is a left -module, where is a subring of generated by its units, (ii) is semisimple artinian if and only if is semiperfect and the matrix ring is a right -ring for all , (iii) Quasi-Frobenius right -rings are Frobenius. [ABSTRACT FROM AUTHOR]

In this note, a necessary and sufficient condition for a trivial extension to be a left p.p.-ring is obtained. As an application, we determine when a trivial Morita context is a left p.p.-ring, and a result of Xue [On p.p. rings, Kobe J. Math. 7 (1990) 77-80] characterizing triangular p.p.-rings follows immediately. [ABSTRACT FROM AUTHOR]

Let M and N be two modules. M is called automorphism N-invariant if for any essential submodule A of N, any essential monomorphism f : A → M can be extended to some g ∈ (N, M). M is called automorphism-invariant if M is automorphism M-invariant. This notion is motivated by automorphism-invariant modules analog discussed in a recent paper by Lee and Zhou [Modules which are invariant under automorphisms of their injective hulls, J. Algebra Appl. 12(2) (2013), 1250159, 9 pp.]. Basic properties of mutually automorphism-invariant modules and automorphism-invariant modules are proved and their connections with pseudo-injective modules are addressed. [ABSTRACT FROM AUTHOR]

For a ring R, there are classical facts that R is right Noetherian if and only if every direct sum of injective right R-modules is injective, and R is right Noetherian if and only if every essential extension of a direct sum of injective hulls of simple right R-modules is a direct sum of injective right R-modules. In this paper, we prove that R is right Noetherian if and only if every essential extension of a direct sum of injective hulls of simple right R-modules is a direct sum of either injective right R-modules or projective right R-modules. [ABSTRACT FROM AUTHOR]

Abstract: Let D be a division algebra finite-dimensional over its center C, , the matrix algebra over D, and V be a vector space over C. We characterize all n-linear forms on Ω in terms of reduced traces and elementary operators. For , it is proved that a bilinear form vanishes on zero products of xy and yx if and only if there exist linear maps such that for all . As an application, a bilinear form B is completely characterized if whenever satisfy , where ξ is a fixed nonzero element in C. [Copyright &y& Elsevier]

Abstract: A ring is called nil-clean if each of its elements is a sum of an idempotent and a nilpotent. In response to a question of S. Breaz et al. in [1], we prove that the matrix ring over a division ring D is a nil-clean ring if and only if . As consequences, it is shown that the matrix ring over a strongly regular ring R is a nil-clean ring if and only if R is a Boolean ring, and that a semilocal ring R is nil-clean if and only if its Jacobson radical is nil and is a direct product of matrix rings over . [Copyright &y& Elsevier]

Let R be a commutative Noetherian ring with non-zero identity, 픞 an ideal of R and M a finitely generated R-module. We assume that N is a weakly Laskerian R-module and r is a non-negative integer such that the generalized local cohomology module is weakly Laskerian for all i < r. Then we prove that is also weakly Laskerian and so is finite. Moreover, we show that if s is a non-negative integer such that is weakly Laskerian for all i, j ≥ 0 with i ≤ s, then is weakly Laskerian for all i ≤ s and j ≥ 0. Also, over a Gorenstein local ring R of finite Krull dimension, we study the question when the socle of is weakly Laskerian? [ABSTRACT FROM AUTHOR]

A ring R is called left morphic if, for each a ∈ R, R/Ra ≅ l(a) or equivalently there exists b ∈ R such that Ra = l(b) and l(a) = Rb, where l(a) and l(b) denote the left annihilators of a and b in R, respectively. Motivated by recent work on left morphic rings, we study the rings R satisfying the property that for each 0 ≠ a ∈ R there exists a unique b ∈ R such that Ra = l(b) and l(a) = Rb. These rings are completely determined in this paper. We also completely determine the rings R with the property that for each 0 ≠ a ∈ R there exists a unique b ∈ R such that Ra = l(b). [ABSTRACT FROM AUTHOR]

Let R be a ring, M be a right R-module and S = EndR(M). M is called a quasi-dual module if, for every R-submodule N of M, N is a direct summand of rM(X) where X ⊆ S. In this article, we study and provide several characterizations of this module classes. We show that if M is quasi-dual module, then, for all m ∈ M, rMlS(m) = mR ⊕ K for some submodule K of M. We also show that every quasidual module is a Kasch module and Z(SM) ⊆ Rad(MR). [ABSTRACT FROM AUTHOR]

Let R be a ring, M a right R-module and a hereditary torsion theory in Mod-R with associated torsion functor τ for the ring R. Then M is called τ-supplemented when for every submodule N of M there exists a direct summand K of M such that K ≤ N and N/K is τ-torsion module. In [4], M is called almost τ-torsion if every proper submodule of M is τ-torsion. We present here some properties of these classes of modules and look for answers to the following questions posed by the referee of the paper [4]: (1) Let a module M = M′ ⊕ M″ be a direct sum of a semisimple module M′ and τ-supplemented module M″. Is M τ-supplemented? (2) Can one find a non-stable hereditary torsion theory τ and τ-supplemented modules M′ and M″ such that M′ ⊕ M″ is not τ-supplemented? (3) Can one find a stable hereditary torsion theory τ and a τ-supplemented module M such that M/N is not τ-supplemented for some submodule N of M? (4) Let τ be a non-stable hereditary torsion theory and the module M be a finite direct sum of almost τ-torsion submodules. Is M τ-supplemented? (5) Do you know an example of a torsion theory τ and a τ-supplemented module M with τ-torsion submodule τ(M) such that M/τ(M) is not semisimple? [ABSTRACT FROM AUTHOR]

Let 풦 be the class of all right R-modules that are kernels of nonzero homomorphisms φ:E1→E2 for some pair of indecomposable injective right R-modules E1,E2. In a previous paper, we completely characterized when two direct sums A1⊕⋯⊕An and B1⊕⋯⊕Bm of finitely many modules Ai and Bj in 풦 are isomorphic. Here we consider the case in which there are arbitrarily, possibly infinitely, many Ai and Bj in 풦. In both the finite and the infinite case, the behaviour is very similar to that which occurs if we substitute the class 풦 with the class 풰 of all uniserial right R-modules (a module is uniserial when its lattice of submodules is linearly ordered). [ABSTRACT FROM AUTHOR]