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We prove that a semiring multiplicatively generated by its idempotents is commutative and Boolean, if every idempotent in the semiring has an orthogonal complement. We prove that a semiring additively generated by its idempotents is commutative, if every idempotent in the semiring has an orthogonal complement and all the nilpotents in the semirings are central. We also provide examples that the assumptions on the existence of orthogonal complements of idempotents and the centrality of nilpotents cannot be omitted. [ABSTRACT FROM AUTHOR]

In this paper, we define and explore in-depth the notion of UQ rings by showing their important properties and by comparing their behavior with that of the well-known classes of UU rings and JU rings, respectively. Specifically, among the other established results, we prove that UQ rings are always Dedekind finite (often named directly finite) as well as that, for semipotent rings R, the following equivalence hold: R/J(R) is UQ⇔R is UJ having the property that the set QN(R) of quasinilpotent elements of R coincides with the Jacobson radical J(R) of R. [ABSTRACT FROM AUTHOR]

A finite semigroup is finitely related (has finite degree) if its term functions are determined by a finite set of finitary relations. For example, it is known that all nilpotent semigroups are finitely related. A nilpotent monoid is a nilpotent semigroup with adjoined identity. We show that every 4-nilpotent monoid is finitely related. We also give an example of a 5-nilpotent monoid that is not finitely related. To our knowledge, this is the first example of a finitely related semigroup where adjoining an identity yields a semigroup which is not finitely related. We also provide examples of finitely related semigroups which have subsemigroups, homomorphic images, and in particular Rees quotients, that are not finitely related. [ABSTRACT FROM AUTHOR]

Suppose that A and G are finite groups such that A acts coprimely on G by automorphisms, we first prove that if every maximal A-invariant subgroup of G that contains the normalizer of some A-invariant Sylow subgroup has index a prime-power and the projective special linear group P S L 2 (7) is not a composition factor of G, then G is solvable. Moreover, we prove that if every non-nilpotent maximal A-invariant subgroup of G has index a prime-power and P S L 2 (7) is not a composition factor of G, then G is solvable. [ABSTRACT FROM AUTHOR]

A subgroup H of a group G is said to be an I C Φ -subgroup of G if H ∩ [ H , G ] ≤ Φ (H). We analyze the structure of a finite group G under the assumption that some given subgroups of G are I C Φ -subgroups of G. A new characterization of finite abelian groups and some new criteria for 2 -nilpotence and nilpotence of finite groups will be obtained. Moreover, we will obtain two criteria for a finite group to lie in a given solvably saturated formation containing the class of finite supersolvable groups. [ABSTRACT FROM AUTHOR]

We prove that the distance from an n × n complex matrix M to the set of nilpotents is at least 1 2 sec ⁡ π n + 2 if there is a nonzero projection P such that P M P = M and M ⁎ M ≥ P. In the particular case where M equals P , this verifies a conjecture by G.W. MacDonald in 1995. We also confirm a related conjecture in D.A. Herrero's book. [ABSTRACT FROM AUTHOR]

The reversible property of rings was introduced by Cohn and has important generalizations in noncommutative ring theory. In this paper, reversibility of rings is investigated in relation with quasinilpotents and idempotents, and our argument is spread out based on this. We call a ring RQnil e-reversible if for any a , b ∈ R , being a ⁢ b = 0 implies b ⁢ a ⁢ e ∈ R qnil for a prescribed idempotent e ∈ R , where R qnil denotes the set of all quasinilpotent elements of R. In the first, we determine the set R qnil for some classes of rings to investigate the structure of Qnil e-reversible rings. In the second, we use R qnil to define Qnil e-reversibility of rings. The notion of Qnil e-reversible ring is a proper generalization of that of e-semicommutative ring, Qnil-semicommutative ring, e-reversible ring and right (left) quasi-duo ring. We obtain some relations between a ring and its quotient rings in terms of Qnil e-reversibility. Applications via some ring extensions and examples illustrating our results are provided. [ABSTRACT FROM AUTHOR]

In this paper, we consider n -dimensional nilpotent finite-dimensional evolution algebras with non-maximal index of nilpotency. We give complete invariants of these algebras in order to classify them. Then, we describe their associative enveloping algebras. We also give a positive answer to the question whether a derivation of this nilpotent evolution algebra is possible to be extended to its enveloping algebra. [ABSTRACT FROM AUTHOR]

Given a simple undirected graph, one can construct from it a c-step nilpotent Lie algebra for every c ≥ 2 and over any field K, in particular also over the real and complex numbers. These Lie algebras form an important class of examples in geometry and algebra, and it is interesting to link their properties to the defining graph. In this paper, we classify the isomorphism classes of K-forms in these real and complex Lie algebras for any subfield K ⊂ C from the structure of the graph. As an application, we show that the number of rational forms up to isomorphism is always one or infinite, with the former being true if and only if the group of graph automorphisms is generated by transpositions. [ABSTRACT FROM AUTHOR]

For n ≥ 2 and fixed k ≥ 1 , we study when an endomorphism f of F n , where F is an arbitrary field, can be decomposed as t + m where t is a root of the unity endomorphism and m is a nilpotent endomorphism with m k = 0. For fields of prime characteristic, we show that this decomposition holds as soon as the characteristic polynomial of f is algebraic over its base field and the rank of f is at least n k , and we present several examples that show that the decomposition does not hold in general. Furthermore, we completely solve this decomposition problem for k = 2 and nilpotent endomorphisms over arbitrary fields (even over division rings). This somewhat continues our recent publications in Linear Multilinear Algebra (2022) and Int. J. Algebra Comput. (2022) as well as it strengthens results due to Cǎlugǎreanu-Lam in J. Algebra Appl. (2016). [ABSTRACT FROM AUTHOR]

In this paper, we study the global dynamics for a class of continuous piecewise ℤ 2 -equivariant cubic Hamiltonian vector fields with nilpotent bicenters at (± 1 , 0). We consider these polynomial vector fields with a challenging case where the bicenters (± 1 , 0) come from the combination of two nilpotent cusps separated by y = 0. We call it a cusp–cusp type. We use the Poincaré compactification, the blow-up theory, the index theory and the theory of discriminant sequence for determining the number of distinct or negative real roots of a polynomial, to classify the global phase portraits of these vector fields in the Poincaré disc. [ABSTRACT FROM AUTHOR]

In this paper, we provide a classification of congruence-simple semirings with a multiplicatively absorbing element and without nontrivial nilpotent elements. [ABSTRACT FROM AUTHOR]

This paper investigates the tensor product of a finite dimensional nilpotent evolution algebra. Some properties that translate from tensor products to factors and vice versa have been investigated, including the index of nilpotency and annihilator. The index of nilpotency of the tensor product of two nilpotent evolution algebras with different indexes of nilpotency is determined. Moreover, we investigate the tensorially decomposable of the 4-dimensional nilpotent evolution algebra. In addition, the decomposable nilpotent evolution algebra with the maximal index of nilpotency has been carried out. [ABSTRACT FROM AUTHOR]

In this paper, we show that every finite-dimensional Zinbiel algebra over an arbitrary field is nilpotent, extending a previous result that they are solvable by other authors. [ABSTRACT FROM AUTHOR]

Let G be a finite group and p a fixed prime divisor of | G |. We prove that if every maximal subgroup of G is nilpotent, or normal, or has p ′ -order, then (1) G is solvable; (2) G has a Sylow tower; (3) there exists at most one prime divisor q ≠ p of | G | such that G is neither q -nilpotent nor q -closed. [ABSTRACT FROM AUTHOR]

We prove a number of results on the survival of the type-I property under extensions of locally compact groups: (a) that given a closed normal embedding \mathbb {N}\trianglelefteq \mathbb {E} of locally compact groups and a twisted action (\alpha,\tau) thereof on a (post)liminal C^*-algebra A the twisted crossed product A\rtimes _{\alpha,\tau }\mathbb {E} is again (post)liminal and (b) a number of converses to the effect that under various conditions a normal, closed, cocompact subgroup \mathbb {N}\trianglelefteq \mathbb {E} is type-I as soon as \mathbb {E} is. This happens for instance if \mathbb {N} is discrete and \mathbb {E} is Lie, or if \mathbb {N} is finitely-generated discrete (with no further restrictions except cocompactness). Examples show that there is not much scope for dropping these conditions. In the same spirit, call a locally compact group \mathbb {G} type-I-preserving if all semidirect products \mathbb {N}\rtimes \mathbb {G} are type-I as soon as \mathbb {N} is, and linearly type-I-preserving if the same conclusion holds for semidirect products V\rtimes \mathbb {G} arising from finite-dimensional \mathbb {G}-representations. We characterize the (linearly) type-I-preserving groups that are (1) discrete-by-compact-Lie, (2) nilpotent, or (3) solvable Lie. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the idempotency, cleanness, and unit-regularity of the commutator [E1,E2] = E1E2 - E2E1 involving idempotents E1,E2 in certain subrings of M2(Z). [ABSTRACT FROM AUTHOR]

A unit-picker is a map that associates to every ring R a well-defined set (R) of central units in R which contains 1 R , is invariant under isomorphisms of rings, is closed under taking inverses, and which satisfies certain set containment conditions for quotient rings, corner rings and matrix rings. Let be a unit-picker. A ring R is called U U if U (R) = (R) + nil (R) and U J if U (R) = (R) + J (R) and U N J if U (R) = (R) + nil (R) + J (R). An extensive study of these rings is conducted, and their connections with strongly nil -clean rings and semi -Boolean rings are investigated. When is specified, known results of U U -rings, U J -rings and U N J -rings are obtained and new results are proved. [ABSTRACT FROM AUTHOR]

Assume that a Lie type algebra admits a Frobenius group of automorphisms with cyclic kernel of order and complement of order such that the fixed-point subalgebra with respect to is trivial and the fixed-point subalgebra with respect to is nilpotent of class . If the ground field contains a primitive th root of unity, then the algebra is nilpotent and the nilpotency class is bounded in terms of and . The result extends the well-known theorem of Khukhro, Makarenko, and Shumyatsky on Lie algebras with metacyclic Frobenius group of automorphisms. [ABSTRACT FROM AUTHOR]

In the paper, we consider n − dimensional naturally graded nilpotent associative algebras with the characteristic sequence C (A) = (n − m , m) over an algebraically closed field of characteristic zero. There are two types of such algebras. We give the algebraic classification up to isomorphism of both types of algebras. Moreover, we prove that there are no such algebras of the second type for n > 2 m + 1 . [ABSTRACT FROM AUTHOR]

We characterize the invertible matrices over a class of semirings such that the set of additively invertible elements is equal to the set of nilpotent elements. We achieve this by studying the liftings of the orthogonal sums of elements that are "almost idempotent" to those that are idempotent. Finally, we show an application of the obtained results to calculate the diameter of the commuting graph of the group of invertible matrices over the semirings in question. [ABSTRACT FROM AUTHOR]

In this paper, we compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional Zinbiel algebras. We study Zinbiel algebras containing maximal abelian subalgebras of codimension 1 and supersolvable Zinbiel algebras in which such subalgebras have codimension 2, and we also analyze the case of filiform Zinbiel algebras. We give examples to clarify some results, including listing the values for α and β for the low dimensional Zinbiel algebras over the complex field that have been classified. Communicated by Alberto Elduque [ABSTRACT FROM AUTHOR]

We obtain a complete characterization of the structure of a finite group G in which every maximal subgroup is nilpotent or a TI-subgroup or has order p ′ for any fixed prime divisor p of | G |. Moreover, we show that there exists at most one prime divisor q of | G | such that G is neither q -nilpotent nor q -closed. [ABSTRACT FROM AUTHOR]

Let G be a finite group. Recently various functions are defined related to the set of order elements of G and using these functions, some interesting criteria for solvability, nilpotency, supersolvability and etc., are obtained. In this paper, we continue this work and for a finite group G, we consider the function m (G) = ∑ g ∈ G 1 / o (g) , where o(g) denotes the order of g ∈ G . In this paper, we prove that if m (G) < m (A 5) , m (G) < m (A 4) , m (G) < m (S 3) , m (G) < m (Q 8) or m (G) < m (C 2 × C 2) , then G is solvable, supersolvable, nilpotent, abelian or cyclic, respectively. [ABSTRACT FROM AUTHOR]

The double extension and the T ⁎ -extension are classical methods for constructing finite dimensional quadratic Lie algebras. The first one gives an inductive classification in characteristic zero, while the latest produces quadratic non-associative algebras (not only Lie) out of arbitrary ones in characteristic different from 2. The classification of quadratic nilpotent Lie algebras can also be reduced to the study of free nilpotent Lie algebras and their invariant forms. In this work we will establish an equivalent characterization among these three construction methods. This equivalence reduces the classification of quadratic 2-step nilpotent to that of trivectors in a natural way. In addition, theoretical results will provide simple rules for switching among them. [ABSTRACT FROM AUTHOR]

Let R   =   A M N B be a Morita context. For generalized fine (respectively, generalized unit-fine) rings A and B , it is proved that R is generalized fine (respectively, generalized unit-fine) if and only if, for a ∈ A and b ∈ B , M b N ⊆ J (A) implies b ∈ J (B) and N a M ⊆ J (B) implies a ∈ J (A). Especially, for fine (respectively, unit-fine) rings A and B , R is fine (respectively, unit-fine) if and only if, for a ∈ A and b ∈ B , M b N = 0 implies b = 0 and N a M = 0 implies a = 0. As consequences, (1) matrix rings over fine (respectively, unit-fine, generalized fine and generalized unit-fine) rings are fine (respectively, unit-fine, generalized fine and generalized unit-fine); (2) a sufficient condition for a simple ring to be fine (respectively, unit-fine) is obtained: a simple ring R is fine (respectively, unit-fine) if both e R e and (1 − e) R (1 − e) are fine (respectively, unit-fine) for some e 2 = e ∈ R ; and (3) a question of Cǎlugǎreanu [1] on unit-fine matrix rings is affirmatively answered. [ABSTRACT FROM AUTHOR]

The aim of this work is to study a very special family of odd-quadratic Lie superalgebras g = g 0 ¯ ⊕ g 1 ¯ such that g 1 ¯ is a weak filiform g 0 ¯ -module (weak filiform type). We introduce this concept after having proved that the unique non-zero odd-quadratic Lie superalgebra (g , B) with g 1 ¯ a filiform g 0 ¯ -module is the abelian 2-dimensional Lie superalgebra g = g 0 ¯ ⊕ g 1 ¯ such that dim g 0 ¯ = dim g 1 ¯ = 1. Let us note that in this context the role of the center of g is crucial. Thus, we obtain an inductive description of odd-quadratic Lie superalgebras of weak filiform type via generalized odd double extensions. Moreover, we obtain the classification, up to isomorphism, for the smallest possible dimensions, that is, six and eight. [ABSTRACT FROM AUTHOR]

In this paper, we continue the study of abelian subalgebras and ideals of maximal dimension for finite-dimensional supersolvable and nilpotent Lie algebras. We show that supersolvable Lie algebras with an abelian subalgebra of codimension 3 contain an abelian ideal with the same dimension, provided that the characteristic of the underlying field is not two, and that the same is true for nilpotent Lie algebras with an abelian subalgebra of codimension 4, provided that the characteristic of the field is greater than five. [ABSTRACT FROM AUTHOR]

For any ring R , let Nil (R) denote the set of nilpotent elements in R , and for any subset S ⊆ R , let S [ x ] denote the set of polynomials with coefficients in S. Due to a celebrated example of Smoktunowicz, there exists a ring R such that Nil (R [ x ]) is a proper subset of Nil (R) [ x ]. In this paper we give an example in the converse direction: there exists a ring R such that Nil (R) [ x ] is a proper subset of Nil (R [ x ]). This is achieved by constructing a ring R with Nil (R) 2 = 0 and a polynomial f ∈ R [ x ] ∖ Nil (R) [ x ] satisfying f 2 = 0. The smallest possible degree of such a polynomial is seven. The example we construct answers an open question of Antoine related to Armendariz rings. [ABSTRACT FROM AUTHOR]

The full transformation semigroup Tn is defined to consist of all functions from Xn = {1, . . ., n} to itself, under the operation of composition. In [9], for any α in Tn, Howie defined and denoted collapse by c(α) = S t∈im(α) {tα-1: |tα-1| ≥ 2}. Let On be the semigroup of all orderpreserving transformations and Cn be the semigroup of all order-preserving and decreasing transformations on Xn under its natural order, respectively. Let E(On) be the set of all idempotent elements of On, E(Cn) and N(Cn) be the sets of all idempotent and nilpotent elements of Cn, respectively. Let U be one of {Cn,N(Cn),E(Cn),On,E(On)}. For α ∈ U, we consider the set imc(α) = {t ∈ im(α): |tα-1| ≥ 2}. For positive integers 2 ≤ k ≤ r ≤ n, we define ... The main objective of this paper is to determine |U(k, r)|, and so |U(k)| for some values r and k. [ABSTRACT FROM AUTHOR]

We prove a structure theorem for homogeneous closed gradient Laplacian solitons and use it to show some examples of closed Laplacian solitons cannot be made gradient. More specifically, we show that the Laplacian solitons on nilpotent Lie groups found by Nicolini are not gradient up to homothetic G 2 -structures except for R 7 , where the potential function must be of a certain form. We also show that one of the closed G 2 -structures constructed by Fernández-Fino-Manero cannot be a gradient soliton. We then examine the structure of almost abelian solvmanifolds admitting closed non-torsion-free gradient Laplacian solitons. [ABSTRACT FROM AUTHOR]

In this paper, we study the global dynamics of continuous piecewise quadratic Hamiltonian systems separated by the straight line x = 0 , where these kinds of systems have a nilpotent center at (0 , 0) , which comes from the combination of two cusps of both Hamiltonian systems. By the Poincaré compactification we classify the global phase portraits of these systems. We must mention that it is extremely rare to find works studying the center-focus problem in piecewise smooth systems with nonelementary singular points as we did here. [ABSTRACT FROM AUTHOR]

We characterize bijective linear maps on M n (ℂ) that preserve the square roots of an idempotent matrix (of any rank). Every such map can be presented as a direct sum of a map preserving involutions and a map preserving square-zero matrices. Next, we consider bijective linear maps that preserve the square roots of a rank-one nilpotent matrix. These maps do not have standard forms when compared to similar linear preserver problems. [ABSTRACT FROM AUTHOR]

P. Hall's classical equality for the number of conjugacy classes in p-groups yields k(G) 1 (3=2) log2 jGj when G is nilpotent. Using only Hall's theorem, this is the best one can do when jGj = 2n. Using a result of G.J. Sherman, we improve the constant 3=2 to 5=3, which is best possible across all nilpotent groups and to 15=8 when G is nilpotent and jGj = 8; 16. These results are then used to prove that k(G) > log3(jGj) when G=N is nilpotent, under natural conditions on N = G. Also, when G is nilpotent of class c, we prove that k(G) (log jGj)t when jGj is large enough, depending only on (e; t). [ABSTRACT FROM AUTHOR]

A Lie superalgebra H satisfying H 2 = Z (H) is called generalized Heisenberg Lie superalgebra. If d (H) := (m ∣ n) is the minimum number of generators required to describe H, then in this article we intend to find the structure of H when precisely dim ⁡ (H 2) = 1 2 [ (m + n) 2 + (n − m) ]. Further, we give some results about the capability, and the Schur multiplier of H. Moreover, we find multiplier M (L) for any nilpotent Lie superalgebra L of nilpotency class 2 with d (L) = (r ∣ s) when its derived subalgebra is of maximum possible dimension and finally show that such an algebra is capable. [ABSTRACT FROM AUTHOR]

Our main result states that if G is a finitely generated soluble group having a normal Abelian subgroup A, such that G/A and are nilpotent (respectively, finite-by-nilpotent, periodic-by-nilpotent, nilpotent-by-finite, finite-by-supersoluble, supersoluble-by-finite) for all (x, a) ∈ G × A, then so is G. We deduce that if 픛 is a subgroup and quotient closed class of groups and if all 2-generated Abelian-by-cyclic groups of 픛 are nilpotent (respectively, finite-by-nilpotent, periodic-by-nilpotent, nilpotent-by-finite, finite-by-supersoluble, supersoluble-by-finite), then so are all finitely generated soluble groups of 픛. We give examples that show that our main result is not true for other classes of groups, like the classes of Abelian, supersoluble, and FC-groups. [ABSTRACT FROM AUTHOR]

We prove the generalized Margulis lemma with a uniform index bound on an Alexandrov n -space X with curvature bounded below, i.e. small loops at p ∈ X generate a subgroup of the fundamental group of the unit ball B 1 (p) that contains a nilpotent subgroup of index ≤ w (n) , where w (n) is a constant depending only on the dimension n. The proof is based on the main ideas of V. Kapovitch, A. Petrunin and W. Tuschmann, and the following results: (1) We prove that any regular almost Lipschitz submersion constructed by Yamaguchi on a collapsed Alexandrov space with curvature bounded below is a Hurewicz fibration. We also prove that such fibration is uniquely determined up to a homotopy equivalence. (2) We give a detailed proof on the gradient push, improving the universal pushing time bound given by V. Kapovitch, A. Petrunin and W. Tuschmann, and justifying in a specific way that the gradient push between regular points can always keep away from extremal subsets. [ABSTRACT FROM AUTHOR]

A unit-picker is a map that associates to every ring R a well-defined set (R) of central units in R which contains 1 R and is invariant under isomorphisms of rings and closed under taking inverses, and which satisfies certain set containment conditions for quotient rings, corner rings and matrix rings. Let be a unit-picker. A ring R is called (strongly) nil -clean if for each a ∈ R , a = v e + b where v ∈ (R) , e 2 = e ∈ R and b ∈ R is nilpotent (and a b = b a). An extensive study of (strongly) nil -clean rings is conducted. When is specified, known results of the much-studied (strongly) nil-clean rings and weakly nil-clean rings are re-proved and new results of other nil-clean-like rings are obtained. [ABSTRACT FROM AUTHOR]

As introduced by Cǎlugǎreanu and Lam in [G. Cǎlugǎreanu and T. Y. Lam, Fine rings: a new class of simple rings, J. Algebra Appl.15(9) (2016) 1650173, 18 pp.], a fine ring is a ring whose every nonzero element is the sum of a unit and a nilpotent. As a natural generalization of fine rings, a ring is called a generalized fine ring if every element not in the Jacobson radical is the sum of a unit and a nilpotent. Here some known results on fine rings are extended to generalized fine rings. A notable result states that matrix rings over generalized fine rings are generalized fine, extending the important result in [G. Cǎlugǎreanu and T. Y. Lam, Fine rings: a new class of simple rings, J. Algebra Appl.15(9) (2016) 1650173, 18 pp.] that matrix rings over fine rings are fine. [ABSTRACT FROM AUTHOR]

We obtain a complete characterization of the structure of a finite group G in which every maximal subgroup is nilpotent or a TI-subgroup or has order p ′ for any fixed prime divisor p of | G |. Moreover, we show that there exists at most one prime divisor q of | G | such that G is neither q -nilpotent nor q -closed. [ABSTRACT FROM AUTHOR]

Let s p 2 n (K) be the symplectic Lie algebra over an algebraically closed field of characteristic zero. We prove that for any nonzero nilpotent element X in s p 2 n (K) there exists a nilpotent element Y in s p 2 n (K) such that X and Y generate s p 2 n (K). [ABSTRACT FROM AUTHOR]

We continue developing the theory of nilpotent symplectic alternating algebras. The algebras of upper bound nilpotent class, that we call maximal algebras, have been introduced and well studied. In this paper, we continue with the external case problem of algebras of minimal nilpotent class. We show the existence of a subclass of algebras over a field F that is of certain lower bound class that depends on the dimension only. This suggests a minimal bound for the class of nilpotent algebras of dimension 2 n ≥ 8 of rank 2 over any field. [ABSTRACT FROM AUTHOR]

In this paper, we begin to study the subalgebra lattice of a Leibniz algebra. In particular, we deal with Leibniz algebras whose subalgebra lattice is modular, upper semi-modular, lower semi-modular, distributive, or dually atomistic. The fact that a non-Lie Leibniz algebra has fewer one-dimensional subalgebras in general results in a number of lattice conditions being weaker than in the Lie case. [ABSTRACT FROM AUTHOR]

We define perfect ideals, near perfect ideals and upper bounded ideals of a finite-dimensional Lie superalgebra, and study the properties of these three kinds of ideals through their relevant sequences. We prove that a Lie superalgebra is solvable if and only if its maximal perfect ideal is zero, or its quotient superalgebra by the maximal perfect ideal is solvable. We also show that a Lie superalgebra is nilpotent if and only if its maximal near perfect ideal is zero. Moreover, we prove that a nilpotent Lie superalgebra has only one upper bounded ideal, which is the nilpotent Lie superalgebra itself. [ABSTRACT FROM AUTHOR]

We give a condition ensuring that the operators in a nilpotent Lie algebra of linear operators on a finite dimensional vector space have a common eigenvector. [ABSTRACT FROM AUTHOR]

In this paper, we introduce Volterra evolution algebras which are evolution algebras whose structural matrices are described by skew symmetric matrices. A main result of the present paper gives a connection between such kind of algebras with ergodicities of Volterra quadratic stochastic operators. Furthermore, some of the properties of the considered algebras such as nilpotency, derivations have been studied as well. [ABSTRACT FROM AUTHOR]

A subalgebra B of a Lie algebra L is called a weak c-ideal of L if there is a subideal C of L such that L = B+C and B∩C ≤ BL where BL is the largest ideal of L contained in B. This is analogous to the concept of weakly c-normal subgroups, which has been studied by a number of authors. We obtain some properties of weak c-ideals and use them to give some characterisations of solvable and supersolvable Lie algebras. We also note that one-dimensional weak c-ideals are c-ideals. [ABSTRACT FROM AUTHOR]

A ∗ -ring R is called a nil ∗ -clean ring if every element of R is a sum of a projection and a nilpotent. Nil ∗ -clean rings are the ∗ -version of nil-clean rings introduced by Diesl. This paper is about the nil ∗ -clean property of rings with emphasis on matrix rings. We show that a ∗ -ring R is nil ∗ -clean if and only if J (R) is nil and R / J (R) is nil ∗ -clean. For a 2-primal ∗ -ring R , with the induced involution given by (a i j ) ∗ = (a i j ∗) T , the nil ∗ -clean property of M n (R) is completely reduced to that of M n (Z 2). Consequently, M n (R) is not a nil ∗ -clean ring for n = 3 , 4 , and M 2 (R) is a nil ∗ -clean ring if and only if J (R) is nil, R / J (R) is a Boolean ring and a ∗ − a ∈ J (R) for all a ∈ R. [ABSTRACT FROM AUTHOR]

Let V be a finite-dimensional vector space over a field of characteristic two. As the main result of this paper, for every nilpotent element e ∈ sl (V) , we describe the Jordan normal form of e on the sl (V) -modules ∧ 2 (V) and S 2 (V). In the case where e is a regular nilpotent element, we are able to give a closed formula. We also consider the closely related problem of describing, for every unipotent element u ∈ SL (V) , the Jordan normal form of u on ∧ 2 (V) and S 2 (V). A recursive formula for the Jordan block sizes of u on ∧ 2 (V) was given by Gow and Laffey [J. Group Theory 9 (2006), 659–672]. We show that their proof can be adapted to give a similar formula for the Jordan block sizes of u on S 2 (V). [ABSTRACT FROM AUTHOR]