University of GuilanJournal of Algebra and Related Topics2345-39316120180601$mathcal{N}$-Fuzzy UP-Algebras and its level subsets124302310.22124/jart.2018.10280.1102ENM. SongsaengUniversity of Phayao, Phayao, ThailanA. IampanUniversity of Phayao, Phayao, Thailand0000-0002-0475-3320Journal Article20180430In this paper, $mathcal{N}$-fuzzy UP-subalgebras (resp., $mathcal{N}$-fuzzy UP-filters, $mathcal{N}$-fuzzy UP-ideals and $mathcal{N}$-fuzzy strongly UP-ideals) of UP-algebras are introduced and considered their generalizations and characteristic $mathcal{N}$-fuzzy sets of UP-subalgebras (resp., UP-filters, UP-ideals and strongly UP-ideals).<br />Further, we discuss the relations between $mathcal{N}$-fuzzy UP-subalgebras (resp., $mathcal{N}$-fuzzy UP-filters, $mathcal{N}$-fuzzy UP-ideals and $mathcal{N}$-fuzzy strongly UP-ideals) and its level subsets.https://jart.guilan.ac.ir/article_3023_eb6a1cfb82810f057804c9773be257d9.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-39316120180601A note on the extended total graph of commutative rings2533302410.22124/jart.2018.10241.1101ENF. Esmaeili Khalil SaraeiUniversity of TehranE. NavidiniaDepartment of Mathematics, University of Guilan, Rasht, IranJournal Article20180425Let $R$ be a commutative ring and $H$ a nonempty proper subset of $R$.<br />In this paper, the extended total graph, denoted by $ET_{H}(R)$ is presented, where $H$ is a<br />multiplicative-prime subset of $R$. It is the graph with all elements of $R$ as vertices, and for distinct $p,qin R$, the vertices $p$ and $q$ are adjacent if and only if $rp+sqin H$ for some $r,sin Rsetminus H$. We also study the two (induced) subgraphs $ET_{H}(H)$ and $ET_{H}(Rsetminus H)$, with vertices $H$ and $Rsetminus H$, respectively. Among other things, the diameter and the girth of $ET_{H}(R)$ are also studied.https://jart.guilan.ac.ir/article_3024_c7a72ce759419a2bcd9ddd57dcf4da67.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-39316120180601Non-reduced rings of small order and their maximal graph3544302510.22124/jart.2018.10130.1097ENA. SharmaDepartment of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, IndiaA. GaurDepartment of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, IndiaJournal Article20180417Let $R$ be a commutative ring with nonzero identity. Let $Gamma(R)$ denotes the maximal graph corresponding to the non-unit elements of R, that is, $Gamma(R)$<br />is a graph with vertices the non-unit elements of $R$, where two distinct<br />vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $R$<br />containing both. In this paper, we investigate that for a given positive integer $n$, is there a non-reduced ring $R$ with $n$ non-units? For $n leq 100$, a complete list of non-reduced decomposable rings $R = prod_{i=1}^{k}R_i$ (up to cardinalities of constituent local rings $R_i$'s) with n non-units is given. We also show that for which $n$, $(1leq n leq 7500)$, $|Center(Gamma(R))|$ attains the bounds in the inequality $1leq |Center(Gamma(R))|leq n$ and for which $n$, $(2leq nleq 100)$, $|Center(Gamma(R))|$ attains the value between the boundshttps://jart.guilan.ac.ir/article_3025_8bced734856b35561fe82af4e15d0d5c.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-39316120180601Tight Closure of a Graded Ideal Relative to a Graded Module4554302610.22124/jart.2018.9589.1092ENF. DorostkarDepartment of Mathematics, University of Guilan, Rasht, IranR. KhosraviDepartment of Mathematics, University of Guilan, Rasht, IranJournal Article20180213In this paper we will study the tight closure of a graded ideal relative to a graded Module.https://jart.guilan.ac.ir/article_3026_973dde807f559bbed7b5db557b368b10.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-39316120180601Essential subhypermodules and their properties5566302710.22124/jart.2018.9573.1089ENB. TalaeeDepartment of Mathematics, Faculty of Basic Sciences, Babol Noshirvani University of Technology, Babol, Iran.Journal Article20180120Let R be a hyperring (in the sense of [8]) andM be a hypermodule on<br /> R. In this paper we will introduce and study a class of subhypermodules<br /> of M. We will study on intersection of this kind of subhypermodules a<br /> give some suitable results about them. We will proceed to give some in-<br /> teresting results about the complements, direct sums and independency<br /> of this kind of subhypermodules.https://jart.guilan.ac.ir/article_3027_30f9c6ce2a292f8d2cb2a3761cca97f0.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-39316120180601Identities in $3$-prime near-rings with left multipliers6777308010.22124/jart.2018.10093.1096ENM. AshrafDepartment of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh 202002, IndiaA. BouaDepartment of Mathematics, Physics and Computer Science, Sidi Mohammed Ben Abdellah University,Taza, MoroccoJournal Article20180409Let $mathcal{N}$ be a $3$-prime near-ring with the center<br />$Z(mathcal{N})$ and $n geq 1$ be a fixed positive integer. In<br />the present paper it is shown that a $3$-prime near-ring<br />$mathcal{N}$ is a commutative ring if and only if it admits a<br />left multiplier $mathcal{F}$ satisfying any one of the following<br />properties: $(i):mathcal{F}^{n}([x, y])in Z(mathcal{N})$, $(ii):mathcal{F}^{n}(xcirc y)in Z(mathcal{N})$,<br />$(iii):mathcal{F}^{n}([x, y])pm(xcirc y)in Z(mathcal{N})$ and $(iv):mathcal{F}^{n}([x, y])pm xcirc yin Z(mathcal{N})$, for all $x, yinmathcal{N}$.https://jart.guilan.ac.ir/article_3080_2bfb17a33939beaf9b8352a63a5aa703.pdf