Journal of Algebra and Related Topics
Volume:9 Issue: 2, Autumn 2021

The aim of this paper is to study the concept of semi-topological UP-algebras which is a UP-algebra supplied with a certain type of topology that makes the binary operation defined on it semi-topologically continuous. This concept is a generalization of the concept of topological UP-algebra. We obtain several properties of semi-topological UP-algebras . Furthermore, we introduce and study the concepts of semi(resp., s, irresolute)-topological UP-homomorphisms and some topological structures on certain types of UP-algebras.

We show that, every derivation of a Lie algebra bundle can be viewed as an inner derivation if one embeds the Lie algebra bundle into a larger Lie algebra bundle. We define, the radical and the nilradical bundle of a Lie algebra bundle and prove, both are characteristic ideal bundles of Lie algebra bundle.

An associative ring R, not necessarily commutative and not necessarily with identity, is called Boolean-zero square orBZS if every element of R is either idempotent or nilpotent of index 2. We continue our investigation of the structure of  niteBZS rings.

In the preprint of ``Pseudo-Gorenstein and Level Hibi Rings,'' Ene, Herzog, Hibi, and Saeedi Madani assert (Theorem 4.3) that for a regular planar lattice $L$ with poset of join-irreducibles $P$, the following are equivalent:(1) $L$ is level;(2) for all $x,yin P$ such that $ylessdot x$, $height_{hat P}(x)+depth_{hat P}(y)lerank(hat P)+1$;(3) for all $x,yin P$ such that $ylessdot x$, either $depth(y)=depth(x)+1$ or $height(x)=height(y)+1$.They added, ``Computational evidence leads us to conjecture that the equivalent conditions given in Theorem 4.3 do hold for any planar lattice (without any regularity assumption).''Ene {sl et al.} prove the equivalence of (2) and (3) for a regular simple planar lattice, and write, ``One may wonder whether the regularity condition ... is really needed.''We show one cannot drop the regularity condition. Ene {sl et al.} say that ``we expect'' (2) to imply (1) for any finite distributive lattice $L$.We provide a counter-example.

Let R be a noncommutative ring. The purpose of this note is to investigate the concept of 2-absorbing (resp., weakly 2-absorbing) p right primary ideals generalizing 2-absorbing (resp., weakly 2-absorbing) ideals of noncommutative rings. From Birkenmeier we have that for commutative rings the notions of primary rings (ideals) coincide with p right primary rings (ideals). Hence all results about 2-absorbing and weakly 2-absorbing primary ideals for commutative rings will follow as special cases from results proved in this note. A number of results concerning 2-absorbing (resp., weakly 2-absorbing) p right primary ideals are given, generalizing the corresponding results from commutative rings to noncommutative rings.

Let $R$ be a commutative ring with identity and let M be an (left) unitary $R$-module. In this paper, we introduce a detail and study the concept of e-small compressible as a generalization of the compressible module, and give some of their properties, characterizations, and examples. On the other hand, we study the relations between e-small compressible modules and some classes of modules.

It is known that all subvarieties of the variety of all semigroups are not absolutely closed. So, we determine some closed homotypical varieties of semigroups determined by the identities  $axy=x^2ayx$, $axy=xa^2ya$, $axy=yay^2x$, $axy=xaya^2$, $axy=y^2ayx$ and $axy=xayx^2$.

In this paper, the authors have defined the valuation near ring. They have proved sometheorem, for example, they have shown every valuation near ring is a local near ring and the ideals of N are totally ordered by inclusion. Also,the symbol valuation $N$-group in near rings has been introduced. Finally, every valuation $N$-groupis a valuation near ring.

Let $D$ be a division ring such that char$(D) neq 2$  and $alpha,beta:Drightarrow D$ be automorphisms of $D$. The main purpose of this paper is to characterizes additive maps                              $f$ and $g$ satisfying the identity $f(x)alpha(x^{-1}) + beta(x)g(x^{-1}) = 0$ for all $0 neq xin D.$ As an application, we describe the structure of an additive map $f$ satisfying the identity                            $f(x)alpha(y)+beta(x)f(y) =l$ for all $x,yin D$ such that $xy=a,$ where $l,ain D$ and $a$ is nonzero. With this, many known results can be either generalized or deduced. In particular, we generalized the results proved incite{C1} and cite{C2}, respectively.

In this paper, we discuss some properties of poset $P$ determined by semi-ideal based zero-divisor graph $G_K(P),$ for a semi-ideal $K$ of $P.$ We investigate some interesting properties if $G_K(P)$ contains a cycle. Further, we prove that if $V(G_K(P))$ is a generalized tree, then $(V(G_K(P))backslash S_x)cup K$ and $(V(G_K(P))backslash S_{geq x})cup K$ are prime semi-ideals of $P$.

In this article, we introduce generalized prime ideal factorization for all proper submodules of a finitely generated module over a Noetherian ring. We show that the generalized prime ideal factorization of a product of two coprime ideals is the product of the generalized prime ideal factorization of the ideals. We find conditions under which the generalized prime ideal factorization of a product of prime ideals is equal to the product of the prime ideals. We show that if R is a Dedekind domain, the generalized prime ideal factorization of an ideal I in R is exactly the prime ideal factorization of I.

Let $(F_n)_{ngeq 0}$ be the Fibonacci sequence given by $F_0 = 0, F_1 = 1$ and $F_{n+2} = F_{n+1}+F_n$ for $n geq 0$. In this paper, we solve all powers of two which are sums of four Fibonacci numbers with a few exceptions that we characterize.