Pairwise commuting derivations of polynomial rings

We prove that n pairwise commuting derivations of the polynomial ring (or the power series ring) in n variables over a field k of characteristic 0 form a commutative basis of derivations if and only if they are k-linearly independent and have no common Darboux polynomials. This result generalizes a recent result due to Petravchuk and is an analogue of a well-known fact that a set of pairwise commuting linear operators on a finite dimensional vector space over an algebraically closed field has a common eigenvector.     

Constants of derivations in polynomial rings over unique factorization domains

A well-known theorem, due to Nagata and Nowicki, states that the ring of constants of any -derivation of , where is a commutative field of characteristic zero, is a polynomial ring in one variable over . In this paper we give an elementary proof of this theorem and show that it remains true if we replace by any unique factorization domain of characteristic zero.

     

On the kernels of some higher derivations in polynomial rings

Let A=R[x₁,…,x_n] be the polynomial ring in n variables over an integral domain R with unit, let D be a rational higher R-derivation on A and let be the extension of D to the quotient field of A. We prove that, if the transcendental degree of the kernel of D over R is not less than n−1, then the quotient field of the kernel of D equals the kernel of . Moreover, when n=2, we give a necessary and sufficient condition for an R-subalgebra of A to be expressed as the kernel of a rational higher R-derivation on A.     

Additive maps on rings behaving like derivations at idempotent-product elements

For every ring R with the unit I containing a nontrivial idempotent P, we describe the additive maps δ from R into itself which behave like derivations, and show that derivations on such kinds of rings can be determined by the action on the elements A,B∈R with AB=0, AB=P and AB=I respectively. Those results of An and Hou [R. An, J. Hou, Characterizations of derivations on triangular rings: additive maps derivable at idempotents, Linear Algebra Appl. 431 (2009) 1070-1080], Bres?ar [M. Bres?ar, Characterizing homomorphisms, multipliers and derivations in rings with idempotents, Proc. Roy. Soc. Edinburgh. Sect. A. 137 (2007) 9-21] and Chebotar et al. [M.A. Chebotar, W.-F. Ke, P.-H. Lee, Maps characterized by action on zero products, Pacific J. Math. 216 (2) 2004 217-228] are improved.     

Commutativity of rings with derivations

I. N. Herstein [10] proved that a prime ring of characteristic not two with a nonzero derivation d satisfying d(x)d(y) = d(y)d(x) for all x, y must be commutative, and H. E. Bell and M. N. Daif [8] showed that a prime ring of arbitrary characteristic with nonzero derivation d satisfying d(xy) = d(yx) for all x, y in some nonzero ideal must also be commutative. For semiprime rings, we show that an inner derivation satisfying the condition of Bell and Daif on a nonzero ideal must be zero on that ideal, and for rings with identity, we generalize all three results to conditions on derivations of powers and powers of derivations. For example, let R be a prime ring with identity and nonzero derivation d, and let m and n be positive integers such that, when charR is finite, m ∨ n < charR. If d(x ^m y ⁿ ) = d(y ⁿ x ^m ) for all x, y ∈ R, then R is commutative. If, in addition, charR≠ 2 and the identity is in the image of an ideal I under d, then d(x) ^m d(y) ⁿ = d(y) ⁿ d(x) ^m for all x, y ∈ I also implies that R is commutative.     

Quotient rings of polynomial rings

Let R be a commutative ring with identity. The multiplicatively closed sets U₂={fR[X]: c(f)–1=R}, (U₂)={fU₂: f is regular} and S={fR[X]: c(f)=R} are studied. By considering various equalities between these sets, many characterizations of Noetherian rings are found. In particular, a Noetherian ring R has depth 1 if and only if S=(U₂): and each maximal ideal of a Noetherian ring is regular if and only if U₂=(U₂).The theory of Prüfer v-multiplication rings (PVMR's) is developed for rings with zero divisors. Six equivalent conditions are given to the statement that an additively regular v-ring R is a PVMR.     

On the derivations of semigroup rings

In questo lavoro studiamo i generatori del modulo delle derivazioni di anelli di semigruppoR=k[S], dandone una caratterizzazione sotto opportune ipotesi suS.     

Local derivations of full matrix rings

Let \({\mathcal{R}}\) be a unital commutative ring and \({\mathcal{M}}\) be a 2-torsion free central \({\mathcal{R}}\) -bimodule. In this paper, for \({n \geqq 3}\), we show that every local derivation from M _n (\({\mathcal{R}}\)) into M _n (\({\mathcal{M}}\)) is a derivation.     

Positive derivations on archimedean lattice-ordered rings

It is known that the only positive derivation on a reduced archimedean f-ring is the zero derivation. We investigate derivations on general archimedean lattice-ordered rings. First, we consider semigroup rings over cyclic semigroups and show that, in the finite case, the only derivation that is zero on the underlying ring is the zero derivation and that, in the infinite case, such derivations are always based on the derivative. Turning our attention to lattice-ordered rings, we show that, on many algebraic extensions of totally ordered rings, the only positive derivation is the zero derivation and that, for transcendental extensions, derivations that are lattice homomorphisms are always translations of the usual derivative and derivations that are orthomorphisms are always dilations of the usual derivative. We also show that the only positive derivation on a lattice-ordered matrix ring over a subfield of the real numbers is the zero derivation, and we prove a similar result for certain lattice-ordered rings with positive squares. The second author thanks Hamilton College for its support of his visits to the first author in Houston. He also thanks John Miller for his friendship and hospitality over the last thirty years.     

Higher derivations and commutativity in lattice-ordered rings

In 1978 I. N. Herstein proved that a prime ring \(R\) of characteristic not two with nonzero derivation \(d\) satisfying \(d(x)d(y)=d(y)d(x)\) for all \(x,y\in R\) is commutative, and in 1995 Bell and Daif showed that \(d(xy)=d(yx)\) implies commutativity. We extend the Bell–Daif theorem to lattice-ordered prime rings with a positive derivation satisfying the property on a one-sided \(L\) -ideal and interpret these conditions for higher derivations in prime \(d\) -rings and in semiprime \(f\) -rings. Our key tool is that every positive derivation nilpotent on a one-sided \(L\) -ideal of a semiprime \(\ell \) -ring is zero on that ideal.     

On derivations and commutativity in prime rings

Supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. A3961.     

Local derivations of subrings of matrix rings

We show that on finite incidence algebras every local derivation is a derivation.