Centrifiers and ring commutativity

A result of Herstein says in particular that if there exists n > 1 such that xᵑ − x ∈ Z(R) for all x in a ring R then R is commutative. We give an elementary proof of this fact for certain values of n, based on the theory of centrifiers which we develop. For n = 5; 7, we also give an elementary proof of the commutativity of rings R such that xᵑ + x ∈ Z(R) for all x ∈ R

Centrifiers and ring commutativity

A result of Herstein says in particular that if there exists n > 1 such that xᵑ − x ∈ Z(R) for all x in a ring R then R is commutative. We give an elementary proof of this fact for certain values of n, based on the theory of centrifiers which we develop. For n = 5; 7, we also give an elementary proof of the commutativity of rings R such that xᵑ + x ∈ Z(R) for all x ∈ R

Centrifiers and ring commutativity

A result of Herstein says in particular that if there exists n > 1 such that xᵑ − x ∈ Z(R) for all x in a ring R then R is commutative. We give an elementary proof of this fact for certain values of n, based on the theory of centrifiers which we develop. For n = 5; 7, we also give an elementary proof of the commutativity of rings R such that xᵑ + x ∈ Z(R) for all x ∈ R

Centrifiers and ring commutativity

A result of Herstein says in particular that if there exists n > 1 such that xᵑ − x ∈ Z(R) for all x in a ring R then R is commutative. We give an elementary proof of this fact for certain values of n, based on the theory of centrifiers which we develop. For n = 5; 7, we also give an elementary proof of the commutativity of rings R such that xᵑ + x ∈ Z(R) for all x ∈ R

Variations on a Theme: Rings Satisfying x³ = x Are Commutative

A ring satisfying x³ = x is necessarily commutative. We consider a variety of weaker forms of this condition and show that many but not all of them imply commutativity. We also present a variety of elementary proofs of the fact that x³ = x implies commutativity

Diurnal variation of adrenocortical activity in chronic fatigue syndrome

Baseline morning and evening serum cortisol and ACTH concentrations, and diurnal changes in hormone levels, were measured in 30 patients with chronic fatigue syndrome (CFS) but without concurrent depressive disorder and a control group of 15 weight-, age- and sex-matched healthy volunteers. Morning cortisol levels were non-significantly lower in CFS patients, while evening levels were non-significantly higher. ACTH concentrations were non-significantly higher in both the morning and evening. The diurnal change in cortisol levels was significantly less in CFS than in controls (p < 0.05). In CFS subjects, evening levels of cortisol correlated significantly with measures of general health and physical functioning, while diurnal change in cortisol was positively correlated with measures of functional improvement over the past year and current social functioning. These results suggest that there is a relationship between adrenocortical function and disability in CFS, but do not reveal the causal connection