Nonzero Nilpotent Elements Research Articles

Modules with reduced endomorphism rings

In this paper, we study endo-reduced modules as modules whose endomorphism rings have no nonzero nilpotent elements. We characterize their properties for different classes of modules, including [Formula: see text]-non-singular modules, multiplication modules and finitely generated modules over commutative Dedekind domains. In the subcategory of finitely generated modules, it is shown that the class of rings [Formula: see text] for which every faithful multiplication [Formula: see text]-module is endo-reduced is precisely that of reduced rings; while the class of rings [Formula: see text] for which every multiplication [Formula: see text]-module is endo-reduced is precisely that of von Neumann regular rings. Characterizations of when an endo-reduced module will be a reduced module are given. We prove that a finitely generated module over a principal ideal domain (PID) is endo-reduced exactly if it is either a semisimple module with pair-wise non-isomorphic submodules or a torsion-free module which is isomorphic to the underlying ring.

Some results on nil-injective rings

Let R be a ring. A right R-module is called nil-injective if for any element w is belong to the set of nilpotent elements, and any right R-homomorphism can be extended to R to M. If RR is nil-injective, then R is called a right nil-injective ring. A right R-module is called Wnil-injective if for each non-zero nilpotent element w of R, there exists a positive integer n such that wn not zero that right R-homomorphism f:wnR to M can be extended to R to M. If RR is right Wnil-injective, then is called a right Wnil-injective ring. In the present work, we discuss some characterizations and properties of right nil-injective and Wnil-injective rings.

Realizing orders as group rings

An order is a commutative ring that as an abelian group is finitely generated and free. A commutative ring is reduced if it has no non-zero nilpotent elements. In this paper we use a new tool, namely, the fact that every reduced order has a universal grading, to answer questions about realizing orders as group rings. In particular, we address the Isomorphism Problem for group rings in the case where the ring is a reduced order. We prove that any non-zero reduced order R can be written as a group ring in a unique “maximal” way, up to isomorphism. More precisely, there exist a ring A and a finite abelian group G, both uniquely determined up to isomorphism, such that R≅A[G] as rings, and such that if B is a ring and H is a group, then R≅B[H] as rings if and only if there is a finite abelian group J such that B≅A[J] as rings and J×H≅G as groups. Computing A and G for given R can be done by means of an algorithm that is not quite polynomial-time. We also give a description of the automorphism group of R in terms of A and G.

Nilpotent elements in the cohomology of the classifying space of a connected Lie group

We give an example of a compact connected Lie group of the lowest rank such that the mod 2 cohomology ring of its classifying space has a nonzero nilpotent element.

Indeks Harmonik dan Indeks Gutman Graf Nilradikal pada Gelanggang Komutatif dengan Satuan

Graph theory is a topic that is still an important subject to discuss. This is because until now graph theory has many practical applications in various disciplines, for example in biology, computer science, economics, informatics engineering, linguistics, mathematics, health, and social sciences. This study discusses the Gutman index of nilradical graphs in the commutative ring with unity.A nilradical graph whose vertices are non-zero nilpotent elements, when the domain is a commutative ring with units, it forms a complete graph only if the commutative ring with units we use is limited to a positive integer modulo n (Z_n). Where n is the square of the prime number p which is less than equal to 3. It is known that the general pattern of harmonic indices and Gutman indices of nilradical graphs in the commutative ring with unity are H(N(Z_(n=p^2 ) ))=((p-2)^2+(p -2))/2(p-2) and Gut(N(Z_(n=p^2 ) ))=(1/2 ((p-2)^2+(p-2))) (p- 2)^2 respectively. In its application, this general form can be used as a numerical parameter of a graph in chemical graph theory, molecular topology, and mathematical chemistry.

On the weakly nilpotent graph of a commutative semiring

Let S be a commutative semiring with unity. In this paper, we introduce the weakly nilpotent graph of a commutative semiring. The weakly nilpotent graph of S, denoted by Γw(S) is defined as an undirected simple graph whose vertices are S and two distinct vertices x and y are adjacent if and only if xy 2 N(S), where S= Sn f0g and N(S) is the set of all non-zero nilpotent elements of S. In this paper, we determine the diameter of weakly nilpotent graph of an Artinian semiring. We prove that if w(S) is a forest, then Γw(S) is a union of a star and some isolated vertices. We study the clique number, the chromatic number and the independence number of Γw(S). Among other results, we show that for an Artinian semiring S, Γw(S) is not a disjoint union of cycles or a unicyclic graph. For Artinian semirings, we determine diam(Γw(S)). Finally, we characterize all commutative semirings S for which Γw(S) is a cycle, where w(S) is the complement of the weakly nilpotent graph of S. Finally, we characterize all commutative semirings S for which Γw(S) is a cycle.

INTEGRABILITY OF CLASSICAL AFFINE W-ALGEBRAS

We prove that all classical affine W-algebras \U0001d4b2(\U0001d524; f), where g is a simple Lie algebra and f is its non-zero nilpotent element, admit an integrable hierarchy of bi-Hamiltonian PDEs, except possibly for one nilpotent conjugacy class in G2, one in F4, and five in E8.

Rickart gamma rings

We introduce the concepts of Rickart gamma ring as a generalized of a Rickart ring and Baer gamma ring. Also, For an Γ —ring R, we show that: (1) R is a right Rickart if and only if it is right p.p.. (2) R is a Baer, quasi-Baer if and only if R is Rickart, p.q. Baer (respectively) and R = {eΓR|e 2 = e ∈ R|, under inclusion, is complete lattice. (3) R is prime if and only if it is quasi-(or right p.q.) Baer and semicentral reduced. (4) If R is Rickart then: (a) R is both right and left nonsingular. (b) R has no nonzero central nilpotent elements. (c) The image isomorphic of R is Rickart. (d) If R is reduced then the idempotent element which is generated the right annihilator of any element in is R unique. (5) The direct product Π i∈I Ri of Γ —rings is Rickart, quasi-(right p.q. ) Baer iff Ri is Rickart quasi-(right p.q.) Baer for all i ∈ I, respectively. (6) The corner and the center of Quasi- Baer, p.q. Baer and Rickart are Quasi-Baer and Baer, p.q. and Rickart and, by conditions, Rickart and Rickart, respectively. (7) We give some conditions to show that (a) If R is Rickart, Quasi-Baer then R Baer and p.q.- Baer, Rickart, respectively. (b) If R is Rickart, then the following statements of R are equivalent: reduced, abelian, idempotents of R commute, the set of idempotents is closed under multiplication, R is commutative at 0, RFI(x) = LFI(x) for every x ∈ R and rR (x) = lR (x) for all x ∈ R.

On nilpotent generators of the symplectic Lie algebra

Let be the symplectic Lie algebra over an algebraically closed field of characteristic zero. We prove that for any nonzero nilpotent element X in there exists a nilpotent element Y in such that X and Y generate .

A Characterization of Left Regularity

We show that a zero-symmetric near-ring $N$ is left regular if and only if $N $ is regular and isomorphic to a subdirect product of integral near-rings, where each component is either an Anshel-Clay near-ring or a trivial integral near-ring. We also show that a zero-symmetric near-ring is regular without nonzero nilpotent elements if and only if the multiplicative semigroup of N is a union of disjoint groups.

The Weakly Nilpotent Graph of a Commutative Ring

AbstractLet R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

Nilradicals of the unique product monoid rings

Armendariz rings are generalization of reduced rings, and therefore, the set of nilpotent elements plays an important role in this class of rings. There are many examples of rings with nonzero nilpotent elements which are Armendariz. Observing structure of the set of all nilpotent elements in the class of Armendariz rings, Antoine introduced the notion of nil-Armendariz rings as a generalization, which are connected to the famous question of Amitsur of whether or not a polynomial ring over a nil coefficient ring is nil. Given an associative ring [Formula: see text] and a monoid [Formula: see text], we introduce and study a class of Armendariz-like rings defined by using the properties of upper and lower nilradicals of the monoid ring [Formula: see text]. The logical relationship between these and other significant classes of Armendariz-like rings are explicated with several examples. These new classes of rings provide the appropriate setting for obtaining results on radicals of the monoid rings of unique product monoids and also can be used to construct new classes of nil-Armendariz rings. We also classify, which of the standard nilpotence properties on polynomial rings pass to monoid rings. As a consequence, we extend and unify several known results.

Annihilators in zero-divisor graphs of semilattices and reduced commutative semigroups

Let V(G) be the set of vertices of a simple connected graph G. The set L1(G) consisting of ∅, V(G), and all neighborhoods N(v) of vertices v∈V(G) is a subposet of the complete lattice L(G) (under inclusion) of all intersections of elements in L1(G). In this paper, it is shown that L1(G) is a join-semilattice and L(G) is a Boolean algebra if and only if G is realizable as the zero-divisor graph of a meet-semilattice with 0. Also, if L1(G) is a meet-semilattice and L(G) is a Boolean algebra, then G is realizable as the zero-divisor graph of a join-semilattice with 0. As a corollary, graphs that are realizable as zero-divisor graphs of commutative semigroups with 0 that do not have any nonzero nilpotent elements are classified.

Fracpairs and fractions over a reduced commutative ring

In the well-known construction of the field of fractions of an integral domain, division by zero is excluded. We introduce “fracpairs” as pairs subject to laws consistent with the use of the pair as a fraction, but do not exclude denominators to be zero. We investigate fracpairs over a reduced commutative ring (that is, a commutative ring that has no nonzero nilpotent elements) and provide these with natural definitions for addition, multiplication, and additive and multiplicative inverse. We find that modulo a simple congruence these fracpairs constitute a “common meadow”, which is a commutative monoid both for addition and multiplication, extended with a weak additive inverse, a multiplicative inverse except for zero, and an additional element a that is the image of the multiplicative inverse on zero and that propagates through all operations. Considering a as an error-value supports the intuition.The equivalence classes of fracpairs thus obtained are called common cancellation fractions (cc-fractions), and cc-fractions over the integers constitute a homomorphic pre-image of the common meadow Qa, the field Q of rational numbers expanded with an a-totalized inverse. Moreover, the initial common meadow is isomorphic to the initial algebra of cc-fractions over the integer numbers. Next, we define canonical term algebras (and therewith normal forms) for cc-fractions over the integers and some meadows that model the rational numbers expanded with a totalized inverse, and we provide some negative results concerning their associated term rewriting properties. Then we consider reduced commutative rings in which the sum of two squares plus one cannot be a zero divisor: by extending the equivalence relation on fracpairs we obtain an initial algebra that is isomorphic to Qa. Finally, we express some negative conjectures concerning alternative specifications for these (concrete) datatypes.

FINITE HIGHER COMMUTATORS IN ASSOCIATIVE RINGS

AbstractIf $T$ is any finite higher commutator in an associative ring $R$, for example, $T= [[R, R] , [R, R] ] $, and if $T$ has minimal cardinality so that the ideal generated by $T$ is infinite, then $T$ is in the centre of $R$ and ${T}^{2} = 0$. Also, if $T$ is any finite, higher commutator containing no nonzero nilpotent element then $T$ generates a finite ideal.

P-STRONGLY REGULAR NEAR-RINGS

Abstract. In this paper we introduce the notion of P-strongly regu-lar near-ring. We have shown that a zero-symmetric near-ring N is P-strongly regular if and only if N is P-regular and P is a completelysemiprime ideal. We have also shown that in a P-strongly regular near-ring N, the following holds: (i) Na+ P is an ideal of N for any a∈N.(ii) Every P-prime ideal of N containing P is maximal. (iii) Every idealI of N fulfills I+ P = I 2 + P. 1. IntroductionThroughout this paper, N denotes a zero-symmetric right near-ring. Aright N-subgroup (left N-subgroup) of N is a subgroup I of (N,+) such thatIN ⊆ I(NI ⊆ I). A quasi-ideal of N is a subgroup Q of (N,+) such thatQN ∩ NQ ⊆ Q. Right N-subgroups and left N-subgroups are quasi-ideals.The intersection of a family of quasi-ideals is again a quasi-ideal.Nis called regular, if for every element aof Nthere exists an element x∈ Nsuch that a= axa. Let P be an ideal of N. Then the near-ring N is said tobe a P-regular near-ring if for each a ∈ N, there exists an element x ∈ Nsuch that a= axa+ p for some p∈ P. If P = 0, then a P-regular near-ringis a regular near-ring. Here the notion of P-regularity is a generalization ofregularity. There are near-rings which are P-regular but not regular.V. A. Andrunakievich [1] defined P-regular rings and S. J. Choi [3] extendedthe P-regularity of a ring to the P-regularity of a near-ring. In this paper weintroduce the notion of P-strongly regular near-ring and obtain equivalent con-ditions for a near-ring to be P-strongly regular. We also introduce the notionsof P-prime ideals and P-near-ring in this paper. I. Yakabe [7] characterizedregular zero-symmetricnear-rings without non-zero nilpotent elements in termsof quasi-ideals. In this paper we characterize P-strongly regular near-ring interms of quasi-ideals. For the basic terminology and notation we refer to [6].

SKEW POLYNOMIAL RINGS OVER SEMIPRIME RINGS

Y. Hirano introduced the concept of a quasi-Armendariz which extends both rings and semiprime rings. A R is called quasi-Armendariz if aiRbj = 0 for each i,j whenever polynomials f(x) = Pm=0 aix i ,g(x) = Pn=0 bjx j 2 R(x) satisfy f(x)R(x)g(x) = 0. In this paper, we first extend the quasi-Armendariz property of semiprime rings to the skew polynomial rings, that is, we show that if R is a semiprime with an epimorphism ae, then f(x)R(x;ae)g(x) = 0 im- plies aiRae i+k (bj) = 0 for any integer k ‚ 0 and i,j, where f(x) = Pm i=0 aix i ,g(x) = Pn j=0 bjx j 2 R(x;ae). Moreover, we extend this prop- erty to the skew monoid rings, the Ore extensions of several types, and skew power series ring, etc. Next we define ae-skew quasi-Armendariz rings for an endomorphism ae of a R. Then we study several exten- sions of ae-skew quasi-Armendariz rings which extend known results for quasi-Armendariz rings and ae-skew rings. Throughout this paper R denotes an associative with identity. We denote by R(x) the polynomial with an indeterminate x over R. Rege and Chhawchharia (18) introduced the notion of an ring. A R is called if whenever polynomials f(x) = Pm=0 aix i ,g(x) = Pn j=0 bjx j 2 R(x) satisfy f(x)g(x) = 0, then aibj = 0 for each i,j. The name Armendariz ring was chosen from the fact that (2, Lemma 1) had showed that a reduced (i.e., a without nonzero nilpotent elements) sat- isfies this condition. Many properties of rings have been studied by several authors (1, 8, 10, 11, 12). Hirano (5) introduced a quasi-Armendariz which is generalizing an ring. A R is called quasi-Armendariz if whenever polynomials f(x) = P m=0 aix i ,g(x) = P n=0 bjx j 2 R(x) satisfy f(x)R(x)g(x) = 0, then aiRbj = 0 for each i,j. Hirano (5, Corollary 3.8) proved that semiprime rings are quasi-Armendariz rings. Moreover, he showed that the class of quasi-Armendariz rings is Morita stable (4, Theorem 3.12 and Proposition 3.13), and that if R is a quasi-Armendariz ring, then some exten- sions of R (e.g., the n-by-n upper triangular matrix ring, the polynomial ring) are also quasi-Armendariz rings. But most of these properties are not stable in rings (for example, (10, Examples 1 and 3, etc.)).

SKEW LAURENT POLYNOMIAL EXTENSIONS OF BAER AND P.P.-RINGS

Let R be a ring and fi a monomorphism of R. We study the skew Laurent polynomial rings R(x,x i1 ;fi) over an fi-skew Armendariz ring R. We show that, if R is an fi-skew Armendariz ring, then R is a Baer (resp. p.p.-)ring if and only if R(x,x i1 ;fi) is a Baer (resp. p.p.-) ring. Consequently, if R is an Armendariz ring, then R is a Baer (resp. p.p.-)ring if and only if R(x,xi1) is a Baer (resp. p.p.-)ring. Throughout this paper R denotes an associative ring with unity and fi : R ! R is an endomorphism, which is not assumed to be surjective. We denote R(x;fi) the Ore extension whose elements are the polynomials § n=0rix i , ri 2 R, where the addition is defined as usual and the multiplication subject to the relation xa = fi(a)x for any a 2 R. The set {x j }j‚0 is easily seen to be a left Ore subset of R(x ;fi), so that one can localize R(x;fi) and form the skew Laurent polynomial ring R(x,x i1 ;fi). Elements of R(x,x i1 ;fi) are finite sums of elements of the form x ij rx i , where r 2 R and i and j are nonnegative integers. A ring R is called Armendariz if whenever polynomials f(x) = a0+a1x+···+ anx n , g(x) = b0+b1x+···+bmx m 2 R(x) satisfy f(x)g(x) = 0, then aibj = 0 for each i,j. The term Armendariz was introduced by Rege and Chhawchharia (20). This nomenclature was used by them since it was Armendariz (2, Lemma 1) who initially showed that a reduced ring (i.e., a ring without nonzero nilpotent elements) always satisfies this condition. According to Krempa (16), an endomorphism fi of a ring R is called to be rigid if afi(a) = 0 implies a = 0 for a 2 R. A ring R is called fi-rigid if there exists a rigid endomorphism fi of R. Note that any rigid endomorphism of a ring is a monomorphism and fi-rigid rings are reduced rings by Hong et al. (10).

On prime rings with commuting nilpotent elements

Let R be a prime ring in which the nilpotent elements commute. If R has finite right uniform dimension or its maximal right quotient ring is Dedekind finite, then R contains no nonzero nilpotent elements.

Idempotent and Nilpotent Submodules of Multiplication Modules

All rings are commutative with identity, and all modules are unital. The purpose of this article is to investigate multiplication von Neumann regular modules. For this reason we introduce the concept of nilpotent submodules generalizing nilpotent ideals and then prove that a faithful multiplication module is von Neumann regular if and only if it has no nonzero nilpotent elements and its Krull dimension is zero. We also give a new characterization for the radical of a submodule of a multiplication module and show in particular that the radical of any submodule of a Noetherian multiplication module is a finite intersection of prime submodules.