Some algebraic applications of graded categorical group theory.
Some characterizations of regular modules.
Let M be a left module over a ring R. M is called a Zelmanowitz-regular module if for each x ∈ M there exists a homomorphism F: M → R such that f(x) = x. Let Q be a left R-module and h: Q → M a homomorphism. We call h locally split if for every x ∈ M there exists a homomorphism g: M → Q such that h(g(x)) = x. M is called locally projective if every epimorphism onto M is locally split. We prove that the following conditions are equivalent:(1) M is Zelmanowitz-regular.(2) every homomorphism into M...
Some conditions on fixed rings.
Some correspondences for Galois skew group rings.
Some graded radicals of graded rings.
Some maximal graduation classes and their Riemannian properties
Some Primitive Differential Operator Rings.
Some properties of prime near-rings with -derivation.
Some remarks on normal property of graded subalgebras of polynomial rings.
Some results for generalized Lie ideals in prime rings with derivation. II.
Some Results on Rings with Chain Conditions.
s-rings and modular representations.
Stable Clifford theory for divisorially graded rings.
Stably free, projective right ideals
Stiff derivations of commutative rings
Strong commutativity preserving maps on Lie ideals of semiprime rings.
Strong stably finite rings and some extensions.
Strongly 2-nil-clean rings with involutions
A -ring is strongly 2-nil--clean if every element in is the sum of two projections and a nilpotent that commute. Fundamental properties of such -rings are obtained. We prove that a -ring is strongly 2-nil--clean if and only if for all , is strongly nil--clean, if and only if for any there exists a -tripotent such that is nilpotent and , if and only if is a strongly -clean SN ring, if and only if is abelian, is nil and is -tripotent. Furthermore, we explore the structure...
Strongly graded left FTF rings.
An associated ring R with identity is said to be a left FTF ring when the class of the submodules of flat left R-modules is closed under injective hulls and direct products. We prove (Theorem 3.5) that a strongly graded ring R by a locally finite group G is FTF if and only if Re is left FTF, where e is a neutral element of G. This provides new examples of left FTF rings. Some consequences of this Theorem are given.
Strongly groupoid graded rings and cohomology
We interpret the collection of invertible bimodules as a groupoid and call it the Picard groupoid. We use this groupoid to generalize the classical construction of crossed products to what we call groupoid crossed products, and show that these coincide with the class of strongly groupoid graded rings. We then use groupoid crossed products to obtain a generalization from the group graded situation to the groupoid graded case of the bijection from a second cohomology group, defined by the grading...