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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]

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 virtually extending (or CIS) if every complement submodule of M is isomorphic to a direct summand of M, and M is called a virtually C2-module if every complement submodule of M which is isomorphic to a direct summand of M is itself a direct summand. The class of virtually extending modules (respectively, virtually C2-modules) is a strict and simultaneous generalization of extending modules (respectively, unifies extending modules and C2-modules): M is a semisimple module if and only if M is virtually semisimple and C2, and M is an extending module if and only if M is virtually extending and virtually C2. Furthermore, every virtually simple right R-module is injective if and only if R is a right V-ring and the class of virtually simple right R-modules coincides with the class of simple right R-modules. Among other results, we show that (1) if all cyclic sub-factors of a cyclic weakly co-Hopfian right R-module M are virtually extending, then M is a finite direct sum of uniform submodules; (2) every distributive virtually extending module over any Noetherian ring is a direct sum of uniform submodules; (3) over a right Noetherian ring, every virtually extending module satisfies the Schröder-Bernstein property; (4) being virtually extending (VC2) is a Morita invariant property; (4) if M⊕E(M) is a VC2-module where E(−) denotes the injective hull, then M is injective. [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]

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]

The aim of the present paper is to introduce and study the dual concepts of weakly automorphism invariant modules and essential tightness. These notions are non-trivial generalizations of both weakly projectivity, dual automorphism invariant property and cotightness. We obtain certain relations between weakly projective modules, weakly dual automorphism invariant modules and superfluous cotight modules. It is proved that: (1) for right perfect rings, every module is a direct summand of a weakly dual automorphism invariant module and (2) weakly dual automorphism invariant modules are precisely superfluous cotight modules. [ABSTRACT FROM AUTHOR]

In this paper, we give a general method of the construction of a 3-dimensional associative algebra R over an arbitrary field F that is a sum of two subalgebras R1 and R2 (i.e. R = R1 + R2). [ABSTRACT FROM AUTHOR]