Characterization and axiomatization of all semigroups whose square is group M.H. Hooshmand Shiraz Branch, Islamic Azad University author text article 2015 eng In this paper we give a characterization for all semigroups whose square is a group. Moreover, we axiomatize such semigroups and study some relations between the class of these semigroups and Grouplikes,introduced by the author. Also, we observe that this paper characterizes and axiomatizes a class of Homogroups (semigroups containing an ideal subgroup).  Finally, several equivalent conditions for a semigroup $S$ with $S^2\leq S$ (the square-group property) will be considered. Algebraic Structures and Their Applications Yazd University 2382-9761 2 v. 2 no. 2015 1 8 http://as.yazd.ac.ir/article_741_50a5e5f483c3aa4d91f526deacc2e032.pdf When does the complement of the annihilating-ideal graph of a commutative ring admit a cut vertex? S. VISWESWARAN Saurashtra University, Rajkot, India author A. PARMAR Saurashtra University, Rajkot, India author text article 2015 eng  The rings considered in this article are  commutative  with identity which admit at least two  nonzero annihilating ideals. Let $R$ be a ring. Let $\mathbb{A}(R)$ denote the set of all annihilating ideals of $R$ and let $\mathbb{A}(R)^{*} = \mathbb{A}(R)\backslash \{(0)\}$. The annihilating-ideal graph of $R$, denoted by $\mathbb{AG}(R)$  is an undirected simple graph whose vertex set is $\mathbb{A}(R)^{*}$ and distinct vertices $I, J$ are joined by an edge in this graph if and only if $IJ = (0)$. The aim of this article is to classify rings  $R$ such that $(\mathbb{AG}(R))^{c}$ ( that is,  the complement of $\mathbb{AG}(R)$)   is connected and admits a cut vertex. Algebraic Structures and Their Applications Yazd University 2382-9761 2 v. 2 no. 2015 9 22 http://as.yazd.ac.ir/article_765_b8befa609c45c0b6b6a79bc456253b4a.pdf Ultra and Involution Ideals in $BCK$-algebras Simin Saidi Goraghani Farhangian University author R. A. Borzooei Shahid Beheshti University author text article 2015 eng In this paper, we define the notions of ultra and involution ideals in $BCK$-algebras. Then we get the relation among them and other ideals as (positive) implicative, associative, commutative and prime ideals. Specially, we show that in a bounded implicative $BCK$-algebra, any involution ideal is a positive implicative ideal and in a bounded positive implicative lower $BCK$-semilattice, the notions of prime ideals and ultra ideals are coincide. Algebraic Structures and Their Applications Yazd University 2382-9761 2 v. 2 no. 2015 23 36 http://as.yazd.ac.ir/article_784_59ea8d93f1f07746b0ae002a32a6a389.pdf The structure of a pair of nilpotent Lie algebras Homayoon Arabyani Islamic Azad University author Hadi Hosseini Fadravi Islamic Azad University author text article 2015 eng Assume that $(N,L)$, is a pair of finite dimensional nilpotent Lie algebras, in which $L$ is non-abelian and $N$ is an ideal in $L$ and also $\mathcal{M}(N,L)$ is the Schur multiplier of the pair $(N,L)$. Motivated by characterization of the pairs $(N,L)$ of finite dimensional nilpotent Lie algebras by their Schur multipliers (Arabyani, et al. 2014) we prove some properties of a pair of nilpotent Lie algebras and generalize results for a pair of non-abelian nilpotent Lie algebras. Algebraic Structures and Their Applications Yazd University 2382-9761 2 v. 2 no. 2015 37 47 http://as.yazd.ac.ir/article_785_f8abf078bb44933f3c1b0a1d39b66275.pdf On the nil-clean matrix over a UFD Somayeh Hadjirezaei Vali-e-Asr University of Rafsanjan author Somayeh Karimzadeh Vali-e-Asr University of Rafsanjan author text article 2015 eng  In this paper we characterize all $2\times 2$ idempotent and nilpotent matrices over an integral domain and then we characterize all $2\times 2$ strongly nil-clean matrices over a PID. Also, we determine when a $2\times 2$ matrix  over a UFD is nil-clean. Algebraic Structures and Their Applications Yazd University 2382-9761 2 v. 2 no. 2015 49 55 http://as.yazd.ac.ir/article_803_7a98829c79d5ccc6521ac399e996e7bb.pdf $z^\circ$-filters and related ideals in $C(X)$ Rostam Mohamadian Shahid Chamran University of Ahvaz author text article 2015 eng In this article we introduce the concept of $z^\circ$-filter on a topological space $X$. We study and investigate the behavior of $z^\circ$-filters and compare them  with corresponding ideals, namely, $z^\circ$-ideals of $C(X)$,  the ring of real-valued continuous functions on a completely regular Hausdorff space $X$. It is observed that $X$ is a compact space if and only if every $z^\circ$-filter is ci-fixed. Finally, by using  $z^\circ$-ultrafilters, we prove that any arbitrary product of i-compact spaces is i-compact. Algebraic Structures and Their Applications Yazd University 2382-9761 2 v. 2 no. 2015 57 66 http://as.yazd.ac.ir/article_807_bb25ddc73dfd82df981f87a48bcc5e25.pdf