Nonzero Nil Research Articles

On the mil graph of module over ring

Let $M$ be an unital left module over a ring $R$ with unity. We define an undirected (\textit{nil graphs}) for the module $M$ as a graph whose vertex set is $M^*=M-{0}$ and any two distinct vertices $x$ and $y$, in these graphs, are adjacent if and only if there exist $r \in R$ such that $r^{2} (x+y) = 0$ and $r(x+y)\neq 0$. In this paper, we study the graph's adjacency, diameter, radius, and eulerian and hamiltonian properties. We also defined another nil graph $\Gamma^{*}_{N} (M)$, in which we reduced the vertex set to $N(M^*)$, set of all non-zero nil elements of the module, and keep the adjacency relation same as that of $\Gamma_{N} (M)$. We investigate the adjacency, diameter, radius, eulerian and hamiltonian properties of the graph $\Gamma^{*}_{N} (\mathbb{Z}_{p^n})$ and compare these properties among both the graphs.

ON NI AND QUASI-NI RINGS

Let $R$ be a ring. It is well-known that $R$ is {\it NI} if and only if $\sum_{i=0}^nRa_iR$ is a nil ideal of $R$ whenever a polynomial $\sum_{i=0}^na_ix^i$ is nilpotent, where $x$ is an indeterminate over $R$. We consider a condition which is similar to the preceding one: $\sum_{i=0}^nRa_iR$ contains a nonzero nil ideal of $R$ whenever $\sum_{i=0}^na_ix^i$ over $R$ is nilpotent. A ring will be said to be {\it quasi-NI} if it satisfies this condition. The structure of quasi-NI rings is observed, and various examples are given to situations which raised naturally in the process.

On the Structure of Nil Graph of a Commutative Ring

Let R be a commutative ring and N(R) be the set of all non-zero nil elements of index two. The Nil Graph of R, denoted by Γ_N(R), is an undirected graph with the vertex set ZN(R)*={x ϵ R*| xy ϵ N(R) for some y in R*=R-{0}} and any two vertices of ZN(R)* are adjacent if and only if xy ϵ N(R). In this paper we determine the chromatic number of the nil graph Γ_N (Z_(p^\alpha q)) . Also we study the diameter and girth of Γ_N (Z_(p^\alpha q)).

Jacobson Radicals of Skew Polynomial Rings of Derivation Type

AbstractWe provide necessary and sufficient conditions for a skew polynomial ring of derivation type to be semiprimitive when the base ring has no nonzero nil ideals. This extends existing results on the Jacobson radical of skew polynomial rings of derivation type.

Jacobson radicals of ring extensions

We extend existing results on the Jacobson radical of skew polynomial rings of derivation type when the base ring has no nonzero nil ideals. We then move to the more general situation of algebras with locally nilpotent skew derivations and examine the Jacobson radical of the algebra when the subalgebra of invariants has no nonzero nil ideals.

On Andrunakievich’s chain and Koethe’s problem

In 1969 Andrunakievich asked whether one gets a ring without nonzero nil left ideals from an arbitrary ring R by factoring out the ideal A(R) which is the sum of all nil left ideals of R. Recently, it was shown that this problem is equivalent to Koethe’s problem. In this context one may consider the chain of ideals, which starts with A1(R) = A(R) ⊆ A2(R), where A2(R)/A1(R) = A(R/A1(R)), and extends by repeating this process. We study the properties of this chain and show that, assuming a negative solution of Koethe’s problem, this chain can terminate at any given ordinal number.

Density of Polynomial Maps

AbstractLet R be a dense subring of End(DV), where V is a left vector space over a division ring D. If dimDV = ∞, then the range of any nonzero polynomial ƒ (X1, … , Xm) on R is dense in End(DV). As an application, let R be a prime ring without nonzero nil one-sided ideals and 0 ≠ a ∈ R. If a f (x1, … , xm)n(xi) = 0 for all x1, … , xm ∈ R, where n(xi ) is a positive integer depending on x1, … , xm, then ƒ (X1, … , Xm) is a polynomial identity of R unless R is a finite matrix ring over a finite field.

Quotient Rings and f-Radical Extensions of Rings

By testing quotient rings, we give another viewpoint concerning the relationship between PI and Goldie properties, etc., and f-radical extensions of rings. The main result proved here is as follows: Let R be a prime algebra without nonzero nil right ideals. Suppose that R is f-radical over a subalgebra A, where f(X 1,…, X t ) is a multilinear polynomial, not an identity for p × p matrices in case char R = p > 0. Suppose that f is not power-central valued in R. Then the maximal ring of right (left) quotients of A coincides with that of R. Moreover, R is right Goldie if and only if A is.

GENERALIZED DERIVATIONS WITH NILPOTENT VALUES ON MULTILINEAR POLYNOMIALS

Let $R$ be a prime ring without nonzero nil one-sided ideals. Suppose that $g$ is a generalized derivation of $R$ and that $f(X_{1},\cdots,X_{k})$ is a multilinear polynomial not central-valued on $R$ such that $g(f(x_{1},\cdots,x_{k}))$ is nilpotent for all $x_{1},\cdots,x_{k}$ in some nonzero ideal of $R$. Then $g=0$.

ANNIHILATORS OF POWER VALUES OF DERIVATIONS ON MULTILINEAR POLYNOMIALS

ABSTRACT Let be a prime ring without nonzero nil one-sided ideals and let be a multilinear polynomial not vanishing on . If is a nonzero derivation of and is a nonzero element in such that for all where is an integer, then is central-valued on and will vanish on the center of .

Derivations with Invertible or Nilpotent Values on a Multilinear Polynomial

Let R be a prime ring with no non-zero nil one-sided ideals, d a nonzero derivation on R, and f(X1,...,X t ) a multilinear polynomial not central-valued on R. Suppose d(f(x1,...,x t )) is either invertible or nilpotent for all x1,...,x t in some non-zero ideal of R. Then it is proved that R is either a division ring or the ring of 2 × 2 matrices over a division ring. This theorem is a simultaneous generalization of a number of results proved earlier.

Derivations with Invertible or Nilpotent Values on a Multilinear Polynomial

Let R be a prime ring with no non-zero nil one-sided ideals, d a nonzero derivation on R, and f(X1,...,X t ) a multilinear polynomial not central-valued on R. Suppose d(f(x1,...,x t )) is either invertible or nilpotent for all x1,...,x t in some non-zero ideal of R. Then it is proved that R is either a division ring or the ring of 2 × 2 matrices over a division ring. This theorem is a simultaneous generalization of a number of results proved earlier.

An Engel condition with derivation for left ideals

We generalize a number of results in the literature by proving the following theorem: Let R be a semiprime ring, D a nonzero derivation of R, L a nonzero left ideal of R, and let [x, y] = xy yx. If for some positive integers tooth, ... , tn, and all x E L, the identity [[... [[D(xtO),xtl],xt2], . .],Xtn] -0 holds, then either D(L) = 0 or else the ideal of R generated by D(L) and D(R)L is in the center of R. In particular, when R is a prime ring, R is commutative. In this paper we prove a theorem generalizing several results, principally [20] and [9], which combine derivations with Engel type conditions. Before stating our theorem we discuss the relevant literature. If one defines [x, y]0 = x and [x, y]1 = [x, y] = xy yx, then an Engel condition is a polynomial [x, Y]n+l = [[X, Y]n, y] in noncommuting indeterminates. A commutative ring satisfies any such polynomial, and a nilpotent ring satisfies one if n is sufficiently large. The question of whether a ring is commutative, or nilpotent, if it satisfies an Engel condition goes back to the well known work of Engel on Lie algebras [15, Chapter 2], and has been considered, with various modifications, by many since then (e.g. [2] or [7]). The connection of Engel type conditions and derivations appeared in a well known paper of E. C. Posner [23] which showed that for a nonzero derivation D of a prime ring R, if [D(x),x] is central for all x C R, then R is commutative. This result has led to many others (see [19] for various references), and in particular to a result of J. Vukman [25] showing that if [D(x), X]2 is central for all x E R, a prime ring with char R + 2,3, then again R is commutative. We extended this result [20] by proving that if [D(x), X]n = 0 for all x c I, an ideal of the prime ring R, then R is commutative, and if instead, this Engel type condition holds for all x C U, a Lie ideal of R, then R embeds in M2(F) for F a field with char F 2. Recently, [9] proved that for a left ideal L of a semiprime ring R, either D(L) = 0 or R contains a nonzero central ideal if either: R is 6-torsion free and [D(x), x]2 is central for all x E L; or if [D(x), Xn] is central for all x C L and R is n!-torsion free. The first of these conditions generalized [1, Theorem 3, p. 99], which assumed that [D(x), x] is central for all x E L, with no restriction on torsion. The second, involving powers, is related to both [12], which showed that a prime ring R is commutative if D(Xk) = 0 for all x C R, and to [8], a significant extension of [12], showing that R is commutative if it contains no nonzero nil ideal and [D(xk(x)), Xk(x)]n = 0 on Received by the editors August 2, 1995. 1991 Mathematics Subject Classification. Primary 16W25; Secondary 16N60, 16U80. ?D1997 American Mathematical Society

Nil Polynomials of Prime Rings

Assume thatRis a prime ring without nonzero nil one-sided ideals and thatf(x1,…,xd) is a polynomial in the noncommuting variablesx1,…,xdand with the coefficients in the extended centroidCofR. If for allr1,…,rd∈R, there exists an integern=n(r1,…,rd)≥1, depending onr1,…,rd∈R, such thatf(r1,…,rd)n=0, then eitherf(r1,…,rd)=0 for allr1,…,rd∈RorRis a finite matrix ring over a finite field.

A result on derivations

Let R be a semiprime ring with a derivation d and let U be a Lie ideal of R, a Ef R. Suppose that ad(u)n = 0 for all u G U, where n is a fixed positive integer. Then ad(I) = 0 for I the ideal of R generated by [U, U] and if R is 2-torsion free, then ad(U) = 0. Furthermore, R is a subdirect sum of semiprime homomorphic images RI and R2 with derivations d1 and d2, induced canonically by d, respectively such that ad1(Rf) = 0 and the image of U in R2 is commutative (central if R is 2-torsion free), where -a denotes the image of a in R1. Moreover, if U = R, then ad(R) = 0. This gives Bregar's theorem without the (n 1)!-torsion free assumption on R. In [8] I. N. Herstein proved that if R is a prime ring and d is an inner derivation of R such that d(x)n = 0 for all x E R and n a fixed integer, then d = O. In [6] A. Giambruno and I. N. Herstein extended this result to arbitrary derivations in semiprime rings. In [2] L. Carini and A. Giambruno proved that if R is a prime ring with a derivation d such that d(x)n(x) = 0 for all x E U, a Lie ideal of R, then d(U) = 0 when R has no nonzero nil right ideals, charR 7f 2 and the same conclusion holds when n(x) = n is fixed and R is a 2-torsion free semiprime ring. Using the ideas in [2] and the methods in [5] C. Lanski [11] removed both the bound on the indices of nilpotence and the characteristic assumptions on R. In [1] M. Bresar gave a generalization of the result due to I. N. Herstein and A. Giambruno [6] in another direction. Explicitly, he proved the theorem: Let R be a semiprime ring with a derivation d, a E R. If ad(x)n = 0 for all x E R, where n is a fixed integer, then ad(R) = 0 when R is an (n 1)!-torsion free ring. The present paper is then motivated by Bresar's result and by Lanski's paper [11]. We prove Bresar's result without the assumption of (n 1)!-torsion free on R. In fact, we study the Lie ideal case as given in [11] and then obtain Bresar's result as the corollary to our main result. More precisely, we shall prove the following Main Theorem. Let R be a semiprime ring with a derivation d and let U be a Lie ideal of R, a E R. Suppose that ad(u)n = 0 for all u E U, where n is a fixed integer. Then ad(I) = 0 for I the ideal of R generated by [U, U] and if R is 2-torsion free, then ad(U) = 0. Furthermore, R is a subdirect sum of semiprime homomorphic images R1 and R2 with derivations d1 and d2, induced canonically by d, respectively such that Received by the editors March 28, 1994 and, in revised form, May 9, 1994 and December 9, 1994. 1991 Mathematics Subject Classification. Primary 16W25.

Hypercentral Derivations

Let k be a given nonnegative integer. Assume that R is a ring without nonzero nil two-sided ideals and that delta is a derivation of R with the property that, for any x ∈ R, [δ( x n( x) ), x n( x) ] k = 0 for some integer n( x) ≥ 1. Let U be the left Utumi quotient ring of R. It is proved here that there exists a central idempotent e of U such that, on the direct sum decomposition U = eU⊕(1 − e) U, the derivation δ vanishes identically on eU and the ring (1 − e) U is commutative. In particular, for any noncommutative prime ring R without nonzero nil two-sided ideals, a derivation δ of R satisfying [δ( x( n ( x) ), x( n ( x) )] k = 0, n( x) ≥ 1, for all x ∈ R, must vanish identically on R.

Rings with nil and power centralK-th commutators

We define thek-th commutator forx, y in a ringR inductively as follows: [x,y]1=[x,y]=xy−yx and [x,y] k =[[x,y] k−1, y ]. Assume thatR is a ring without nonzero nil onesided ideals. The following are shown: (1) If [x,y] k is nilpotent for allx,y∈R, thenR must be commutative. (2) If [x,y] k is power central for allx,y∈R, thenR must satisfy the standard polynomial of degree 4.

Onn-th commutators with nilpotent or regular values in rings

Then-th commutator for a,b in a ringR is defined inductively as follows: [a,b]1=[a,b]=ab−ba and[a,b] n=[[a,b]−1,b]. We characterize the ringsR without non-zero nil right ideals in which[a,b] nis nilpotent or regular for alla,b∈R. We also examine the case whereR is a semiprime ring with involution in which[t 1, t2]nis nilpotent or regular for all tracest 1,t2∈R.

ON THE COMMUTATIVITY AND ANTICOMMUTATIVITY OF RINGS II

It is shown that if $R$ is any associative ring such that for each $x,y\in R$, there exist an even natural number $m(x,y)$ and an odd natural number $n(x,y)$, depending on $x$ and $y$, with either $[x,y]^{m(x,y)} = [x,y]^{n(x,y)}$ or $(x\circ y)^{m(x, y)} = (x \circ y)^{n(x,y)}$, then either $[x,y]$ or $(x \circ y)$ is nilpotent for all $x, y$ in $R$. Moreover, $R$ is commutative if $R$ has no nonzero nil right ideals.

Commutativity conditions on rings

We prove the following result: let R be an arbitrary ring with centre Z such that for every x, y ∈ R, there exists a positive integer n = n(x, y) ≥ 1 such that (xy)n − ynxn ∈ Z and (yx)n − xnyn ∈ Z; then, if R has no non-zero nil ideals, R is commutative. We also prove a result on commutativity of general rings: if R is r!-torsion free and for all x, y ∈ R, [xr, ys] = 0 for fixed integers r ≥ s ≥ 1, then R is commutative. As a corollary we obtain that if R is (n + 1)!-torsion free and there exists a fixed n ≥ 1 such that (xy)n − ynxn = (yx)n − xnyn ∈ Z for all x, y ∈ R, then R is commutative.