Characterizations of N(2,2,0) Algebras

The so-called ideal and subalgebra and some additional concepts of N(2,2,0) algebras are discussed. A partial order and congruence relations on N(2,2,0) algebras are also proposed, and some properties are investigated.

On ϕ-Absorbing Primary Elements in Lattice Modules

Let L be a C-lattice and let M be a lattice module over L. Let ϕ:M→M be a function. A proper element P∈M is said to be ϕ-absorbing primary if, for x1,x2,…,xn∈L and N∈M, x1x2⋯xnN≤P and x1x2⋯xnN≰ϕ(P) together imply x1x2⋯xn≤(P:1M) or x1x2⋯xi-1xi+1⋯xnN≤PM, for some i∈{1,2,…,n}. We study some basic properties of ϕ-absorbing primary elements. Also, various generalizations of prime and primary elements in multiplicative lattices and lattice modules as ϕ-absorbing elements and ϕ-absorbing primary elements are unified.

Generalized Malcev-Neumann Series Modules with the Beachy-Blair Condition

We introduce a new class of extension rings called the generalized Malcev-Neumann series ring R((S;σ;τ)) with coefficients in a ring R and exponents in a strictly ordered monoid S which extends the usual construction of Malcev-Neumann series rings. Ouyang et al. in 2014 introduced the modules with the Beachy-Blair condition as follows: A right R-module satisfies the right Beachy-Blair condition if each of its faithful submodules is cofaithful. In this paper, we study the relationship between the right Beachy-Blair condition of a right R-module MR and its Malcev-Neumann series module extension MSR((S;σ;τ)).

Involutions in the Automorphism Groups of Small Sporadic Simple Groups

For each of fifteen of the sporadic finite simple groups we determine the suborbits of its automorphism group in its conjugation action upon its involutions. Representatives are obtained as words in standard generators.

On the Torsion Units of Integral Adjacency Algebras of Finite Association Schemes

Torsion units of group rings have been studied extensively since the 1960s. As association schemes are generalization of groups, it is natural to ask about torsion units of association scheme rings. In this paper we establish some results about torsion units of association scheme rings analogous to basic results for torsion units of group rings.

Zero Divisor Graph for the Ring of Eisenstein Integers Modulo n

Let En be the ring of Eisenstein integers modulo n. In this paper we study the zero divisor graph Γ(En). We find the diameters and girths for such zero divisor graphs and characterize n for which the graph Γ(En) is complete, complete bipartite, bipartite, regular, Eulerian, Hamiltonian, or chordal.

Right k-Weakly Regular Γ-Semirings

The concepts of a k-idempotent Γ-semiring, a right k-weakly regular Γ-semiring, and a right pure k-ideal of a Γ-semiring are introduced. Several characterizations of them are furnished.

Injectivity of the Composition Operators of Étale Mappings

Let X be a topological space. The semigroup of all the étale mappings of X (the local homeomorphisms X→X) is denoted by et(X). If G∈et(X), then the G-right (left) composition operator on et(X) is defined by RG LG:et(X)→et(X), RGF=F∘G (LGF=G∘F). When are the composition operators injective? The Problem originated in a new approach to study étale polynomial mappings C2→C2 and in particular the two-dimensional Jacobian conjecture. This approach constructs a fractal structure on the semigroup of the (normalized) Keller mappings and outlines a new method of a possible attack on this open problem (in preparation). The construction uses the left composition operator and the injectivity problem is essential. In this paper we will completely solve the injectivity problems of the two composition operators for (normalized) Keller mappings. We will also solve the much easier surjectivity problem of these composition operators.

Finitely Generated Modules over Group Rings of a Direct Product of Two Cyclic Groups

Let K be a commutative field of characteristic p>0 and let G=G1×G2, where G1 and G2 are two finite cyclic groups. We give some structure results of finitely generated K[G]-modules in the case where the order of G is divisible by p. Extensions of modules are also investigated. Based on these extensions and in the same previous case, we show that K[G]-modules satisfying some conditions have a fairly simple form.

Residuation Properties and Weakly Primary Elements in Lattice Modules

We obtain some elementary residuation properties in lattice modules and obtain a relation between a weakly primary element in a lattice module M and weakly prime element of a multiplicative lattice L.