On generalized commutators and nil commutator ideals

Archiv der Mathematik -     

Rings whose singular ideals are nil

Abstract

It is well known that when a ring R satisfies ACC on right annihilators of elements, then the right singular ideal of R is nil, in this case, we say R is right nil-singular. Many classes of rings whose singular ideals are nil, but do not satisfy the ACC on right annihilators, are presented and the behavior of them is investigated with respect to various constructions, in particular skew polynomial rings and triangular matrix rings. The class of right nil-singular rings contains π-regular rings and is closed under direct sums. Examples are provided to explain and delimit our results.     

Vanishing power values of commutators with derivations

Let R be a prime ring of char R ≠ 2, let d be a nonzero derivation of R, and let ρ be a nonzero right ideal of R such that [[d(x)x ⁿ , d(y)] _m , [y, x] _s ] ^t = 0 for all x, y ? ρ, where n ≥ 1, m ≥ 0, s ≥ 0, and t ≥ 1 are fixed integers. If [ρ, ρ]ρ ≠ 0 then d(ρ)ρ = 0.     

The nil radical of power series rings

We describe the nil radical of power series rings in non-commuting indeterminates by showing that a series belongs to the radical if and only if the ideal generated by its coefficients is nilpotent. We also show thatt the principal ideals generated by elements of the nil radical of the power series ring in one indeterminate are nil of bounded index.     

A note on nil power serieswise Armendariz rings

A ring R is called nil power serieswise Armendariz if $ \forall f = \sum\limits_{i = 0}^\infty {a_i X^i } $ \forall f = \sum\limits_{i = 0}^\infty {a_i X^i } and $ g = \sum\limits_{i = 0}^\infty {b_i X^i } $ g = \sum\limits_{i = 0}^\infty {b_i X^i } in R[[X]] such that f g ∈ Nil(R)[[X]], then a _i b _j ∈ Nil(R) for all i and j. In this note we characterize completely nil power serieswise Armendariz rings with their nilradical Nil(R) (where the nilradical is the set of nilpotent elements). We prove that a ring is nil power serieswise Armendariz if and only if Nil(R) is an ideal of R. We prove that each power serieswise Armendariz ring is nil power serieswise Armendariz and we give examples of nil power serieswise Armendariz rings.     

Rings of formal power series with homeomorphic prime spectra

LetR?T be domains, not fields, such that Spec(R)=Spec(T) as sets; that is, such that the prime ideals ofT coincide, as sets, with those ofR. It is proved that the canonical map Spec(T[[X]])→Spec(R[[X]]) is a homeomorphism. This generalizes a result of Girolami in caseR is a pseudovaluation domain with the SFT (strong finite type)—property andT is its associated valuation domain. The analogous property for polynomial rings is also characterized: Spec(T[X])→Spec(R[X]) is a homeomorphism if and only ifR/M?T/M is a purely inseparable (algebraic) field extension, whereM is the maximal ideal ofR.     

Operator semigroups with quasinilpotent commutators

It is shown that a multiplicative semigroup of operators is triangularizable if is quasinilpotent for every pair in the semigroup and certain other hypotheses are satisfied.

     

Centers with degenerate infinity and their commutators

We study polynomial systems with degeneracy at infinity and a center-focus equilibrium at the origin. We give some general properties related to the existence of polynomial commutators and use these properties in order to characterize uniformly isochronous polynomial centers with polynomial commutator and, also, we show that the commutator of the centers of the analytic systems whose angular speed is constant can be chosen of radial form. Finally, we characterize the systems (−y+P_s+∑_j=kⁿ⁻¹xH_j,x+Q_s+∑_j=kⁿ⁻¹yH_j)^t with polynomial commutator, with P_j,Q_j,H_j and K_j homogeneous polynomials.     

Rings whose multiplicative endomorphisms are power functions

Let R be a commutative ring. For any positive integer m, the power function f:R→R defined by f(x):=x ^m is easily seen to be an endomorphism of the multiplicative semigroup (R,?). In this note, we characterize the commutative rings R with identity for which every multiplicative endomorphism of (R,?) is equal to a power function. Specifically, we show that every endomorphism of (R,?) is a power function if and only if R is a finite field.     

Derivations and nil ideals

IfR is a torsion free ring, then the nil radical ofR is stable relative to any derivation ofR. In this note, we show the same is true for the Levitzki radical ofR, which is the largest locally nilpotent ideal ofR.     

Equal-time commutators with improved energy-momentum tensor

Theoretical and Mathematical Physics -     

Nonlinear commutators and jacobians

Journal of Fourier Analysis and Applications -