Primes and Right Ideals in Right Cones

A right cone H in a group G is a submonoid of G that generates G and aH ? bH for a, b ? H implies bH ? aH. With any right ideal I ≠ H of H a completely prime ideal P _r (I) of H is associated and the set 𝒫(I) of right ideals I′ of H with the same associated prime ideal P′ =P _r (I) is determined if P′·? P″ is a right invariant segment in H. The set 𝒫(I) is also described if P _r (I) is a limit prime.     

Generalizations of Prime Ideals

Let R be a commutative ring with identity. Various generalizations of prime ideals have been studied. For example, a proper ideal I of R is weakly prime (resp., almost prime) if a, b ∈ R with ab ∈ I ? {0} (resp., ab ∈ I ? I ²) implies a ∈ I or b ∈ I. Let φ:?(R) → ?(R) ∪ {?} be a function where ?(R) is the set of ideals of R. We call a proper ideal I of R a φ-prime ideal if a, b ∈ R with ab ∈ I ? φ(I) implies a ∈ I or b ∈ I. So taking φ_?(J) = ? (resp., φ₀(J) = 0, φ₂(J) = J ²), a φ_?-prime ideal (resp., φ₀-prime ideal, φ₂-prime ideal) is a prime ideal (resp., weakly prime ideal, almost prime ideal). We show that φ-prime ideals enjoy analogs of many of the properties of prime ideals.     

On Generalizations of Prime Ideals (II)

Let R be a commutative ring with identity. We say that a proper ideal P of R is (n ? 1, n)-weakly prime (n ≥ 2) if 0 ≠ a ₁…a _n  ∈ P implies a ₁…a _i?1 a _i+1…a _n  ∈ P for some i ∈ {1,…, n}, where a ₁,…, a _n  ∈ R. In this article, we study (n ? 1, n)-weakly prime ideals. A number of results concerning (n ? 1, n)-weakly prime ideals and examples of (n ? 1, n)-weakly prime ideals are given. Rings with the property that for a positive integer n such that 2 ≤ n ≤ 5, every proper ideal is (n ? 1, n)-weakly prime are characterized. Moreover, it is shown that in some rings, nonzero (n ? 1, n)-weakly prime ideals and (n ? 1, n)-prime ideals coincide.     

Group homomorphisms inducing isomorphisms in cohomology

In this article, we call a ring R right generalized semiregular if for any a ∈ R there exist two left ideals P, L of R such that lr(a) = P⊕ L, where P ? Ra and Ra ∩ L is small in R. The class of generalized semiregular rings contains all semiregular rings and all AP-injective rings. Some properties of these rings are studied and some results about semiregular rings and AP-injective rings are extended. In addition, we call a ring R semi-π-regular if for any a ∈ R there exist a positive integer n and e ² = e ∈ a ⁿ R such that (1 ? e)a ⁿ  ∈ J(R), the Jacobson radical of R. It is shown that a ring R is semi-π-regular if and only if R/J(R) is π-regular and idempotents can be lifted modulo J(R).     

Rings Over which the Krull Dimension and the Noetherian Dimension of All Modules Coincide

We denote by 𝒜(R) the class of all Artinian R-modules and by 𝒩(R) the class of all Noetherian R-modules. It is shown that 𝒜(R) ? 𝒩(R) (𝒩(R) ? 𝒜(R)) if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)), for all centrally prime ideals P (i.e., ab ∈ P, a or b in the center of R, then a ∈ P or b ∈ P). Equivalently, if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)) for all normal prime ideals P of R (i.e., ab ∈ P, a, b normalize R, then a ∈ P or b ∈ P). We observe that finitely embedded modules and Artinian modules coincide over Noetherian duo rings. Consequently, 𝒜(R) ? 𝒩(R) implies that 𝒩(R) = 𝒜(R), where R is a duo ring. For a ring R, we prove that 𝒩(R) = 𝒜(R) if and only if the coincidence in the title occurs. Finally, if Q is the quotient field of a discrete valuation domain R, it is shown that Q is the only R-module which is both α-atomic and β-critical for some ordinals α,β ≥ 1 and in fact α = β = 1.     

On Generalized Stable Ideals

An ideal I of a ring R is generalized stable in case aR + bR = R with a ∈ I, b ∈ R implies that there exist s, t ∈ 1 + I such that s(a + by)t = 1 for a y ∈ R. We establish, in this article, necessary and sufficient conditions for an ideal of a regular ring to be generalized stable. It is shown that every regular square matrix over such ideals admits a diagonal reduction. These extend the corresponding results of generalized stable regular rings.     

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.     

On Generalizations of Prime Ideals

Let R be a commutative ring with identity. Let φ: S(R) → S(R) ∪ {?} be a function, where S(R) is the set of ideals of R. Suppose n ≥ 2 is a positive integer. A nonzero proper ideal I of R is called (n ? 1, n) ? φ-prime if, whenever a ₁, a ₂, ?, a _n  ∈ R and a ₁ a ₂?a _n  ∈ I?φ(I), the product of (n ? 1) of the a _i 's is in I. In this article, we study (n ? 1, n) ? φ-prime ideals (n ≥ 2). A number of results concerning (n ? 1, n) ? φ-prime ideals and examples of (n ? 1, n) ? φ-prime ideals are also given. Finally, rings with the property that for some φ, every proper ideal is (n ? 1, n) ? φ-prime, are characterized.     

Orders in regular rings with minimal condition for principal right ideals

ABSTRACT

A ring R is called an n-clean (resp. Σ-clean) ring if every element in R is n-clean (resp. Σ-clean). Clean rings are 1-clean and hence are Σ-clean. An example shows that there exists a 2-clean ring that is not clean. This shows that Σ-clean rings are a proper generalization of clean rings. The group ring ?_(p) G with G a cyclic group of order 3 is proved to be Σ-clean. The m× m matrix ring M _m (R) over an n-clean ring is n-clean, and the m×m (m>1) matrix ring M _m (R) over any ring is Σ-clean. Additionally, rings satisfying a weakly unit 1-stable range were introduced. Rings satisfying weakly unit 1-stable range are left-right symmetric and are generalizations of abelian π-regular rings, abelian clean rings, and rings satisfying unit 1-stable range. A ring R satisfies a weakly unit 1-stable range if and only if whenever a ₁ R + ˙˙˙ a _m R = dR, with m ≥ 2, a ₁,…, a _m, d ∈ R, there exist u ₁ ∈ U(R) and u ₂,…, u _m ∈ W(R) such that a ₁ u ₁ + ? a _m u _m = Rd.     

Generalized Derivations with Engel Conditions on One-Sided Ideals

Let R be a noncommutative prime ring and I a nonzero left ideal of R. Let g be a generalized derivation of R such that [g(r ^k ), r ^k ] _n  = 0 for all r ∈ I, where k, n are fixed positive integers. Then there exists c ∈ U, the left Utumi quotient ring of R, such that g(x) = xc and I(c ? α) = 0 for a suitable α ∈ C. In particular we have that g(x) = α x, for all x ∈ I.     

On n-Absorbing Ideals of Commutative Rings

Let R be a commutative ring with 1 ≠ 0 and n a positive integer. In this article, we study two generalizations of a prime ideal. A proper ideal I of R is called an n-absorbing (resp., strongly n-absorbing) ideal if whenever x ₁…x _n+1 ∈ I for x ₁,…, x _n+1 ∈ R (resp., I ₁…I _n+1 ? I for ideals I ₁,…, I _n+1 of R), then there are n of the x _i 's (resp., n of the I _i 's) whose product is in I. We investigate n-absorbing and strongly n-absorbing ideals, and we conjecture that these two concepts are equivalent. In particular, we study the stability of n-absorbing ideals with respect to various ring-theoretic constructions and study n-absorbing ideals in several classes of commutative rings. For example, in a Noetherian ring every proper ideal is an n-absorbing ideal for some positive integer n, and in a Prüfer domain, an ideal is an n-absorbing ideal for some positive integer n if and only if it is a product of prime ideals.     

Characterizations of QF Rings

A ring R is called “left GP-injective” if for any 0 ≠ a ∈ R, there exists n > 0 such that a ⁿ  ≠ 0 and a ⁿ R = r(l(a ⁿ )). It is proved that (1) every right Noetherian left GP-injective ring such that every complement left ideal is a left annihilator is a QF ring, (2) every left GP-injective ring with ACC on left annihilators such that every complement left ideal is a left annihilator is a QF ring, and (3) every left P-injective left CS ring satisfying ACC on essential right ideals is a QF ring. Several well-known results on QF rings are obtained as corollaries.     

Mod-Retractable Rings

A right module M over a ring R is said to be retractable if Hom _R (M, N) ≠ 0 for each nonzero submodule N of M. We show that M ? _R RG is a retractable RG-module if and only if M _R is retractable for every finite group G. The ring R is (finitely) mod-retractable if every (finitely generated) right R-module is retractable. Some comparisons between max rings, semiartinian rings, perfect rings, noetherian rings, nonsingular rings, and mod-retractable rings are investigated. In particular, we prove ring-theoretical criteria of right mod-retractability for classes of all commutative, left perfect, and right noetherian rings.     

Primes of Gauss Extensions Over Crossed Product Algebras

Let K be a skew field with total subring V and G be a right ordered group with cone P, so that the crossed product algebra K*G has a skew field D of fractions. We consider total subrings R of D with R ∩ K = V, describe the overrings in D, as well as subrings of R. For particular extensions R of V we determine the prime ideals of R in terms of prime ideals of V and prime ideals of overcones of P in G.     

Split-By-Nilpotent Extensions Algebras and Stratifying Systems

Let Γ and Λ be artin algebras such that Γ is a split-by-nilpotent extension of Λ by a two sided ideal I of Γ. Consider the change of rings functors G: =_ΓΓ_Λ ?_Λ ? and F: =_ΛΛ_Γ ?_Γ ?. In this article, by assuming that I _Λ is projective, we find the necessary and sufficient conditions under which a stratifying system (Θ, ≤) in modΛ can be lifted to a stratifying system (GΘ, ≤) in mod(Γ). Furthermore, by using the functors F and G, we study the relationship between their filtered categories of modules; and some connections with their corresponding standardly stratified algebras are stated (see Theorem 5.12, Theorem 5.15 and Theorem 5.18). Finally, a sufficient condition is given for stratifying systems in mod(Γ) in such a way that they can be restricted, through the functor F, to stratifying systems in mod(Λ).     

The Ordered Group of Invertible Ideals of a Prüfer Domain of Finite Character

Abstract

Let D be a Prüfer domain, and denote by ± b?(D) the multiplicative group of all invertible fractional ideals of D, ordered by A ≤ B if and only if A ? B. Denote by G _i the value group of the valuation associated with the valuation ring D _{M i} , where {M _i } _i∈I is the collection of all maximal ideals of D. In this note we prove that the natural map from ± b?(D) into ± b∏ _i∈I G _i is an isomorphism onto the cardinal sum ± b? _i∈I G _i if and only if D is h-local. As a corollary, the group of divisibility of an h-local Bézout domain is isomorphic to ± b? _i∈I G _i , the notation being as above.     

On Strongly Stable Rings

Abstract

A ring R is called strongly stable if whenever aR + bR = R, there exists a w ∈ Q(R) such that a + bw ∈ U(R), where Q(R) = {x ∈ R ∣ ? e ? e ² ∈ J(R), u ∈ U(R) such that x = eu}. These rings are shown to be a natural generalization of semilocal rings and unit regular rings. We investigate the extensions of strongly stable rings. K ₁-groups of such rings are also studied. In this way we recover and extend some results of Menal and Moncasi.     

Left Annihilators of Commutators with Derivation on Right Ideals

Abstract

Let R be a prime ring of characteristic different from 2, d a non-zero derivation of R, I a non-zero right ideal of R, a ∈ R, S ₄(x ₁,…, x ₄) the standard polynomial in 4 variables. Suppose that, for any x, y ∈ I, a[d([x, y]), [x, y]] = 0. If S ₄(I, I, I, I)I ≠ 0, then aI = ad(I) = 0.     

Localization of Frames

Motivated by Balan, Casazza, Heil, and Landau's results on localization of frames in a separable Hilbert space ?, we investigate the transitivity of localization relation of the frame ? = {f _i } _i∈I and sequence ? = {e _j } _j∈G in ? with respect to the associated map a: I → G, where G is a discrete abelian group and I is a index set. We also study some properties of ? _p -column decay, ? _p -row decay, weak homogeneous approximation property, and strong homogeneous approximation property of frame ? = {f _i } _i∈I , and sequence ? = {e _j } _j∈G , with respect to the associated map a.