2015
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Characterization and axiomatization of all semigroups whose square is group
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2
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 $S^2leq S$ (the squaregroup property) will be considered.
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1
8


M.H.
Hooshmand
Shiraz Branch, Islamic Azad University
Shiraz Branch, Islamic Azad University
Iran
Ideal subgroup
grouplike
homogroup
class united grouplike
real grouplike
[[1] A. H. Cliord and D. D. Miller, Semigroups having zeroid elements, Amer. J. Math. vol. 70 (1948), 117125.##[2] D. P. Dawson, Semigroups Having Left of Right Zeroid Elements, Acta Scientiarum Mathematicarum, XXVII (1966), 93957.##[3] M.H.Hooshmand, Grouplikes, Bull. Iran Math. Soc., Bull. Iran. Math. Soc., vol. 39, no. 1 (2013), 6586.##[4] M.H.Hooshmand and H. Kamarul Haili, Decomposer and Associative Functional Equations, Indag. Mathem., N.S., vol.18, no. 4 (2007), 539554.##[5] M.H.Hooshmand, Upper and Lower Periodic Subsets of Semigroups, Algebra Colloquium, vol. 18, no.3 (2011), 447460.##[6] M.H. Hooshmand and H. Kamarul Haili, Some Algebraic Properties of bParts of Real Numbers, Siauliai Math.Semin., vol.3, no.11 (2008), 115121.##[7] M.H. Hooshmand and S. Rahimian, A study of regular grouplikes, J. Math. Ext., vol. 7, no. 4 (2013), 1{9.##[8] R. P. Hunter, On the structure of homogroups with applications to the theory of compact connected semigroups, Fund. Math. vol. 52 (1963), 69102.##[9] A. Nagy, Special Classes of Semigroups, Kluwer Academic Publishers, 2001.##[10] G. Thierrin, Contribution a la theorie des equivalences dans les demigroupes, Bull. Soc. Math. France, vol. 83 (1955), 103159.##]
When does the complement of the annihilatingideal graph of a commutative ring admit a cut vertex?
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2
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 annihilatingideal 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.
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9
22


S.
VISWESWARAN
Saurashtra University, Rajkot
Saurashtra University, Rajkot
India


A.
PARMAR
Saurashtra University, Rajkot
Saurashtra University, Rajkot
India
Nprime of $(0)$
Bprime of $(0)$
complement of the annihilatingideal graph of a commutative ring
vertex cut and cut vertex of a connected graph
[[1] G Aalipour, S. Akbari, M. Behboodi, R. Nikandish, M.J. Nikmehr, and F. Shaiveisi, The classication of##annihilatingideal graph of commutative rings, Alg. Colloquium, 21, 249 (2014), doi:10.1143/S1005386714000200.##[2] G. Aalipour, S. Akbari, R. Nikandish, M.J. Nikmehr, and F. Shaiveisi, On the coloring of the annihilating##ideal graph of a commutative ring, Discrete Math., 312 (2012), 26202625.##[3] .D.F. Anderson, M.C. Axtell, J.A. Stickles Zerodivisor graphs in commutative rings, in Commutative##Algebra, Noetherian and NonNoetherian perspectives, M. Fontana, S.E. Kabbaj, B. Olberding, and I.##Swanson (Editors), SpringerVerlag, New York, 2011, 2345.##[4] D.F. Anderson and P.S. Livingston, The zerodivisor graph of a commutative ring, J. Alg. 217 (1999),##[5] M.F. Atiyah and I.G. Macdonald, Introduction to commutative algebra, AddisonWesley, Reading, Mas##sachusetts, 1969.##[6] M.C. Axtell, N. Baeth, and J.A. Stickles, Cut vertices in zerodivisor graphs of nite commutative rings,##Comm. Alg., 39(6) (2011), 21792188. ##[7] R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, Universitext, Springer, New york,##[8] I. . Beck, Coloring of commutative rings, J. Alg. 116 (1988), 208226.##[9] M. Behboodi and Z. Rakeei, The annihilatingideal graph of commutative rings I, J. Alg. Appl. 10 (2011),##[10] M. Behboodi and Z. Rakeei, The annihilatingideal graph of commutative rings II, J. Alg. Appl.10 (2011),##[11] B. Cotee, C. Ewing, M. Huhn, C.M. Plaut, and E.D. Weber, CutSets in zerodivisor graphs of nite##commutative rings, Comm. Alg. 39(8) (2011), 28492861.##[12] R. Gilmer and W. Heinzer, The Laskerian property, power series rings and Noetherian spectra, Proc. Amer.##Math. Soc. 79(1) (1980), 1316.##[13] W. Heinzer and J. Ohm, Locally Noetherian commutative rings, Trans. Amer. Math. Soc. 158(2) (1971),##[14] W. Heinzer and J. Ohm, On the Noetherianlike rings of E.G. Evans, Proc. Amer. Math. Soc. 34(1) (1972),##[15] M. Hadian, Unit action and geometric zerodivisor ideal graph, Comm. Alg. 40 (2012), 29202930.##[16] I. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago, 1974.##[17] T. Tamizh Chelvam and K. Selvakumar, On the connectivity of the annihilatingideal graphs, Discuss.##Math. Gen. Alg. Appl. 35 (2015), 195204.##[18] S. Visweswaran, Some results on the complement of the zerodivisor graph of a commutative ring, J. Alg.##Appl. 10(3) (2011), 573595.##[19] S. Visweswaran, Some properties of the complement of the zerodivisor graph of a commutative ring, ISRN Alg. 2011 (2011), Article ID 591041, 24 pages.##[20] S. Visweswaran and Hiren D. Patel, Some results on the complement of the annihilating ideal graph of a##commutative ring, J. Algebra Appl. 14 (2015), doi: 10.1142/S0219498815500991, 23 pages.##[21] S. Visweswaran, When does the complement of the zerodivisor graph of a commutative ring admit a cut##vertex?, Palestine J. Math. 1(2) (2012), 138147.##]
Ultra and Involution Ideals in $BCK$algebras
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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.
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23
36


Simin
Saidi Goraghani
Farhangian University
Farhangian University
Iran
siminsaidi@yahoo.com


R. A.
Borzooei
Shahid Beheshti University
Shahid Beheshti University
Iran
$BCK$algebra
(associative
commutative
positive implicative
implicative) ideal
ultra ideal
involution ideal
[[1] R. A. Borzooei, J. Shohani, Fraction structures on bounded implicative BCKalgebras, Word Academy of##Science, Engineering and Thechnology, 49, 10841090 (2009).##[2] O. HeuboKwegna and J. B. Nganou, A Global Local Principle for BCKmodules, International Journal of##Algebra, 5(14), 691702 (2011).##[3] Y. Huang, BCIalgebra, Science Press, Beijing (2006).##[4] Y. Imai and K. Iseki, On axiom systems of propositional calculi, Proceedings of the Japan Academy, 42,##1921 (1966).##[5] K. Iseki, On ideals in BCKalgebras, Mathematics Seminar Notes, 3, 112 (1975).##[6] K. Iseki and S. Tanaka, Ideal theory of BCKalgebras, Mathematica Japonica, 21, 351366 (1976).##[7] P. Jiayin, Normed BCKalgebras, Advances in Mathematics, 4, 492500 (2011).##[8] J. Meng and Y. B. Jun, BCKalgebras, Kyung Moon Sa Co, seoul, (1994).##[9] Z. M. Samaei and M. A. N. Azadani, A Class of BCKalgebras, International Journal of Algebra, 28, 13791385 (2011).##]
The structure of a pair of nilpotent Lie algebras
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2
Assume that $(N,L)$, is a pair of finite dimensional nilpotent Lie algebras, in which $L$ is nonabelian 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 nonabelian nilpotent Lie algebras.
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37
47


Homayoon
Arabyani
Islamic Azad University
Islamic Azad University
Iran
arabyani_h@yahoo.com


Hadi Hosseini
Fadravi
Islamic Azad University
Islamic Azad University
Iran
Nilpotent Lie algebra
Pair of Lie algebras
Schur multiplier
[[1] J. M. AncocheaBermdez and M. Goze, Classication des algbres de Lie nilpotentes de dimension 7, Arch.##Math., 52(2) (1989), 157185.##[2] H. Arabyani, F. Saeedi, M. R. R. Moghaddam and E. Khamseh, On characterizing a pair of nilpotent Lie##algebras by their Schur multipliers, Comm. Alg, 42 (2014), 54745483.##[3] P. Batten, Multipliers and covers of Lie algebras, PhD thesis, North Carolina State university, (1993).##[4] P. Batten, K. Moneyhum and E. Stitzinger, On characterizing nilpo tent Lie algebras by their multipliers,##Comm. Algebra, 24(14) (1996), 43194330.##[5] P. Batten, E. Stitzinger, On covers of Lie algebras, Comm. Algebra, 24(14), (1996), 43014317.##[6] L. R. Bosko, On Schur multipliers of Lie algebras and groups of maximal class,internat. J. Algebra comput,##20(6), (2010), 807821.##[7] S. Cicalo, W. A. de Graaf and C. Schneider, Six dimensional nilpotent Lie algebras, Linear Algebra Appl.,##436(1) (2012), 163189.##[8] J. Dixmier, Sur les reprsentations unitaires des groupes de Lie nilpo tents III, Canad. J. Math., 10 (1958),##[9] G. Ellis, A nonabelian tensor square of Lie algebras, Glasgow Math. J. (39) (1991), 101{120.##[10] K. Erdmann and M. Wildon, Introduction to Lie Algebras, Springer undergraduate Mathematics series,##[11] P. Hardy, On characterizing nilpotent Lie algebras by their multipliers (III), Comm. Algebra. 33 (2005),##42054210.##[12] P. Hardy and E. Stitzinger, On characterizing nilpotent Lie algebras by their multipliers, t(L) = 3; 4; 5; 6,##Comm. Algebra. 26(11) (1998), 35273539.##[13] M. R. R. Moghaddam, A. R. Salemkar and K. Chiti, Some properties on the Schur multiplier of a pair of##groups, J. Algebra 312 (2007), 18.##[14] M. R. R. Moghaddam, A. R. Salemkar and T. Karimi, Some inequalities for the order of Schur multiplier##of a pair of groups, Comm. Algebra 36 (2008), 16.##[15] K. Moneyhun, Isoclinism in Lie algebras, Algebra Groups Geom., 11 (1994), 922.##[16] V. V. Morozov, Classication des algebras de Lie nilpotents de dimension 6. Izv. Vyssh. Ucheb. Zar, 4##(1958), 161{171.##[17] P. Niroomand, On dimension of the Schur multiplier of nilpotent Lie algebras,cent. Eur. J. Math., 9(1)##(2011), 5764.##[18] P. Niroomand and F.G. Russo, A note on the Schur multiplier of a nilpotent Lie algebra, Comm. Algebra,##39 (2011), 12931297.##[19] M.R. Rismanchian and M. Araskhan, Some inequalities for the dimension of the Schur multilier of a pair##of (nilpotent) Lie algebras, J. Algebra , 352 (2012), 173179.##[20] M. R. Rismanchian and M. Araskhan, Some properties on the Schur multiplier of a pair of Lie algebras,##J. Algebra Appl., 11 (2012), 1250011(9 pages).##[21] D. J. S. Robinson, A Course in the Theory of Groups, Springer Verlag, New York, 1982.##[22] F. Saeedi, A. R. Salemkar and B. Edalatzadeh, The commutator subalgebra and Schur multiplier of a pair##of nilpotent Lie algebras, J. Lie Theory, 21 (2011), 491498.##[23] F. Saeedi, H. Arabyani and P. Niroomand, On dimension of the Schur multiplier of nilpotent Lie algebras##(II),(submitted).##[24] A. R. Salemkar, V. Alamian, H. Mohammadzadeh, Some properties of the Schur multiplier and covers of##Lie algebras, Comm. Algebra. 36(2) (2008), 697707.##[25] A. R. Salemkar and S. Alizadeh Niri, Bounds for the dimension of the Schur multiplier of a pair of nilpotent Lie algebras, AsianEur. J. Math. 5 (2012), 1250059(9 pages).##[26] B. Yankosky, On the multiplier of a Lie algebra, J. Lie Theory, 13(1) (2003), 16.##]
On the nilclean matrix over a UFD
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2
In this paper we characterize all $2times 2$ idempotent and nilpotent matrices over an integral domain and then we characterize all $2times 2$ strongly nilclean matrices over a PID. Also, we determine when a $2times 2$ matrix over a UFD is nilclean.
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49
55


Somayeh
Hadjirezaei
ValieAsr University of Rafsanjan
ValieAsr University of Rafsanjan
Iran
s.hajirezaei@vru.ac.ir


Somayeh
Karimzadeh
ValieAsr University of Rafsanjan
ValieAsr University of Rafsanjan
Iran
Rank of a matrix
Idempotent matrix
Nilpotent matrix
Nilclean matrix
Strongly nilclean matrix
[[1] S. Breaz, G. Calugaranu, P. Danchev, T. Micu, Nilclean matrix rings, Linear Algebra Appl., vol. 439, no. 1 (2013), 31153119.##[2] A. J. Diesl, Classes of strongly clean rings, Phd thesis, University of California, Berkeley, 2006.##[3] A. J. Diesl, Nil clean rings, J. Algebra, vol. 383, (2013), 197211.##[4] T. W. Hungerford, Algebra, SpringerVerlag, 1980.##[5] M.T. Kossan, T. K. Lee, Y. Zhou, When is every matrix over a division ring a sum of an idempotent and a nilpotent, Linear Algebra Appl., vol. 450 (2014), 712.##[6] W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc., vol. 229 (1977), 269278.##[7] G. Song, X. Guo, Diagonability of idempotent matrices over noncommutative rings, Linear Algebra Appl., vol. 297, no. 1 (1999), 17.##]
$z^circ$filters and related ideals in $C(X)$
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2
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 realvalued 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 cifixed. Finally, by using $z^circ$ultrafilters, we prove that any arbitrary product of icompact spaces is icompact.
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57
66


Rostam
Mohamadian
Shahid Chamran University of Ahvaz
Shahid Chamran University of Ahvaz
Iran
mohamadian_r@scu.ac.ir
$z^circ$filter
prime $z^circ$filter
cifree $z^circ$filter
ifree $z^circ$filter
$z^circ$ultrafilter
icompact
[[1] F. Azarpanah, On almost Pspaces, Far East J. Math. Sci., Special Volume, 121{132 (2000).##[2] F. Azarpanah, O.A.S. Karamzadeh and A. Rezaei Aliabad, On zideals in C(X), Fund. Math., 160, 15{25 (1999).##[3] F. Azarpanah, O.A.S. Karamzadeh and A. Rezaei Aliabad, On ideal consisting entirely of zerodivisor, Comm. Algebra, 28(2), 1061{1073 (2000).##[4] F. Azarpanah and M. Karavan, On nonregular ideals and zideal in C(X), Cech. Math. J., 55(130), 397{407 (2005).##[5] F. Azarpanah and R. Mohamadian, pzideals and pzideals in C(X), Acta. Math. Sinica., English Series, 23,##989{996 (2007).##[6] R. Engelking, General Topology, PWNPolish Sci Publ, 1977.##[7] L. Gillman and M. Jerison, Rings of Continuous Functions, SpringerVerlag, New York, 1976.##[8] G. Mason, zideals and prime ideals, J. Algebra, 26: 280{297 (1973).##]