JOURNAL OF ALGEBRAIC SYSTEMS
Year:2019 | Volume:7 | Issue:1 |Papers Count:7

A ring R with an automorphism  and a -derivation  is called -quasi-Baer (resp., -invariant quasi-Baer) if the right annihilator of every -ideal (resp., -invariant ideal) of R is generated by an idempotent, as a right ideal. In this paper, we study Baer and quasi-Baer properties of skew PBW extensions. More exactly, let A =  (R) ⟨ x1; : : :; xn⟩ be a skew PBW extension of derivation type of a ring R. (i) It is shown that R is Δ-quasi-Baer if and only if A is quasi-Baer. (ii) R is Δ-Baer if and only if A is Baer, when R has IFP. Also, let A =  (R) ⟨ x1; : : :; xn⟩ be a quasi-commutative skew PBW extension of a ring R. (iii) If R is a -quasi-Baer ring, then A is a quasi-Baer ring. (iv) If A is a quasi-Baer ring, then R is a -invariant quasi-Baer ring. (v) If R is a -Baer ring, then A is a Baer ring, when R has IFP. (vi) If A is a Baer ring, then R is a -invariant Baer ring. Finally, we show that if A =  (R) ⟨ x1; : : :; xn⟩ is a bijective skew PBW extension of a quasi-Baer ring R, then A is a quasi-Baer ring.

Let 􀀀 be a group, 􀀀 ′ a subgroup of 􀀀 with finite index and M be a 􀀀-module. We show that M is cotorsion if and only if it is cotorsion as a 􀀀 ′-module. Using this result, we prove that the global cotorsion dimensions of rings Z􀀀 and Z􀀀 ′ are equal.

An M-polysymmetrical hyperring (R; +;
) is an algebraic system, where (R; +) is an M-polysymmetrical hypergroup, (R;
) is a semigroup and
is bilaterally distributive over +. We introduce the concept of hyperideals of an M-polysymmetrical hyperring and by using this concept, we construct an ordinary quotient ring. Finally, the fundamental theorem of homomorphism is derived in the context of M-polysymmetrical hyperrings.

Let R be a ring (not necessarily commutative) with nonzero identity. We define 􀀀 (R) to be the graph with vertex set R in which two distinct vertices x and y are adjacent if and only if there exist unit elements u; v of R such that x+uyv is a unit of R. In this paper, basic properties of 􀀀 (R) are studied. We investigate connectivity and the girth of 􀀀 (R), where R is a left Artinian ring. We also determine when the graph 􀀀 (R) is a cycle graph. We prove that if 􀀀 (R)  = 􀀀 (Mn(F)) then R  = Mn(F), where R is a ring and F is a finite field. We show that if R is a finite commutative semisimple ring and S is a commutative ring such that 􀀀 (R)  = 􀀀 (S), then R  = S. Finally, we obtain the spectrum of 􀀀 (R), where R is a finite commutative ring.

For all M; N 2 P(U) such that M  N, we first introduced the definitions of (M; N)-uni-soft ideals and (M; N)-uni-soft interior ideals of an ordered semigroup and studied them. WhenM= ∅ and N = U, we meet the ordinary soft ones. Then we proved that in regular and in intra-regular ordered semigroups the concept of (M; N)-uni-soft ideals and the (M; N)-uni-soft interior ideals coincide. Finally, we introduced (M; N)-uni-soft simple ordered semigroup and characterized the simple ordered semigroups in terms of (M; N)-uni-soft interior ideals.

The purpose of this paper is to introduce new methods for constructing generalized groups, generalized topological groups and top spaces. We study some properties of these structures and present some relative concrete examples. Moreover, we obtain generalized groups by using of Hilbert spaces and tangent spaces of Lie groups, separately.

A t-Cayley hypergraph X = t-Cay(G; S) is called normal for a finite group G, if the right regular representation R(G) of G is normal in the full automorphism group Aut(X) of X. In this paper, we investigate the normality of t-Cayley hypergraphs of abelian groups, where jSj  4.