University of GuilanJournal of Algebra and Related Topics2345-39313220151201A class of J-quasipolar rings1151537ENS. HaliciogluAnkara UniversityM. B. CalciAnkara UniversityA. HarmanciHacettepe UniversityJournal Article20151013In this paper, we introduce a class of $J$-quasipolar rings. Let $R$ be a ring with identity. An element $a$ of a ring $R$ is called {it weakly $J$-quasipolar} if there exists $p^2 = pin comm^2(a)$ such that $a + p$ or $a-p$ are contained in $J(R)$ and the ring $R$ is called {it weakly $J$-quasipolar} if every element of $R$ is weakly $J$-quasipolar. We give many characterizations and investigate general properties of weakly $J$-quasipolar rings. If $R$ is a weakly $J$-quasipolar ring, then we show that (1) $R/J(R)$ is weakly $J$-quasipolar, (2) $R/J(R)$ is commutative, (3) $R/J(R)$ is reduced. We use weakly $J$-quasipolar rings to obtain more results for $J$-quasipolar rings. We prove that the class of weakly $J$-quasipolar rings lies between the class of $J$-quasipolar rings and the class of quasipolar rings. Among others it is shown that a ring $R$ is abelian weakly $J$-quasipolar if and only if $R$ is uniquely clean.https://jart.guilan.ac.ir/article_1537_d40640a41f82ff681c817e78291f88e6.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-39313220151201Omega-almost Boolean rings17251538ENM. Phani Krishna KishoreGVP college of Engineering (Autonomous)Journal Article20150921In this paper the concept of an $Omega$- Almost Boolean ring is introduced and illistrated how a sheaf of algebras can be constructed from an $Omega$- Almost Boolean ring over a locally Boolean space.https://jart.guilan.ac.ir/article_1538_cf62d3cb0228e37d1bdaa5d77771591d.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-39313220151201The total graph of a commutative semiring with respect to proper ideals27411539ENZ. Ebrahimi SarvandiUniversity of GuilanS. Ebrahimi AtaniUniversity of GuilanJournal Article20150709Let $I$ be a proper ideal of a commutative semiring $R$ and let $P(I)$ be the set of all elements of $R$ that are not prime to $I$. In this paper, we investigate the total graph of $R$ with respect to $I$, denoted by $T(Gamma_{I} (R))$. It is the (undirected) graph with elements of $R$ as vertices, and for distinct $x, y in R$, the vertices $x$ and $y$ are adjacent if and only if $x + y in P(I)$. The properties and possible structures of the two (induced) subgraphs $P(Gamma_{I} (R))$ and $bar {P}(Gamma_{I} (R))$ of $T(Gamma_{I} (R))$, with vertices $P(I)$ and $R - P(I)$, respectively are studied.https://jart.guilan.ac.ir/article_1539_c4cf79602c757856bbf5ef810db8ebf5.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-39313220151201Small submodules with respect to an arbitrary submodule43511540ENR. BeyranvandLorestan UniversityF. MoradiLorestan UniversityJournal Article20150929Let $R$ be an arbitrary ring and $T$ be a submodule of an $R$-module $M$. A submodule $N$ of $M$ is called $T$-small in $M$ provided for each submodule $X$ of $M$, $Tsubseteq X+N$ implies that $Tsubseteq X$. We study this mentioned notion which is a generalization of the small submodules and we obtain some related results.https://jart.guilan.ac.ir/article_1540_f62c06857579fe38edbcdc2cf29eabbb.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-39313220151201Castelnuovo-Mumford regularity of products of monomial ideals53591541ENS. YangSoochow UniversityL. ChuSoochow UniversityY. QianSoochow UniversityJournal Article20151027Let $R=k[x_1,x_2,cdots, x_N]$ be a polynomial ring over a field $k$. We prove that for any positive integers $m, n$, $text{reg}(I^mJ^nK)leq mtext{reg}(I)+ntext{reg}(J)+text{reg}(K)$ if $I, J, Ksubseteq R$ are three monomial complete intersections ($I$, $J$, $K$ are not necessarily proper ideals of the polynomial ring $R$), and $I, J$ are of the form $(x_{i_1}^{a_1}, x_{i_2}^{a_2}, cdots, x_{i_l}^{a_l})$.https://jart.guilan.ac.ir/article_1541_3cee72350a59ab590b45b3ebf213a8d1.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-39313220151201nth-roots and n-centrality of finite 2-generator p-groups of nilpotency class 261711542ENM. PolkoueiUniversity of GuilanM. HashemiUniversity of GuilanJournal Article20151201Here we consider all finite non-abelian 2-generator $p$-groups ($p$ an odd prime) of nilpotency class two and study the probability of having $n^{th}$-roots of them. Also we find integers $n$ for which, these groups are $n$-central.https://jart.guilan.ac.ir/article_1542_21a4c2d0eea4260bab01d638f0dd1e22.pdf