ORIGINAL_ARTICLE
Characterization and axiomatization of all semigroups whose square is group
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.
http://as.yazd.ac.ir/article_741_50a5e5f483c3aa4d91f526deacc2e032.pdf
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1
8
Ideal subgroup
grouplike
homogroup
class united grouplike
real grouplike
M.H.
Hooshmand
true
1
Shiraz Branch, Islamic Azad University
Shiraz Branch, Islamic Azad University
Shiraz Branch, Islamic Azad University
LEAD_AUTHOR
[1] A. H. Cliord and D. D. Miller, Semigroups having zeroid elements, Amer. J. Math. vol. 70 (1948), 117-125.
1
[2] D. P. Dawson, Semigroups Having Left of Right Zeroid Elements, Acta Scientiarum Mathematicarum, XXVII (1966), 93-957.
2
[3] M.H.Hooshmand, Grouplikes, Bull. Iran Math. Soc., Bull. Iran. Math. Soc., vol. 39, no. 1 (2013), 65-86.
3
[4] M.H.Hooshmand and H. Kamarul Haili, Decomposer and Associative Functional Equations, Indag. Mathem., N.S., vol.18, no. 4 (2007), 539-554.
4
[5] M.H.Hooshmand, Upper and Lower Periodic Subsets of Semigroups, Algebra Colloquium, vol. 18, no.3 (2011), 447-460.
5
[6] M.H. Hooshmand and H. Kamarul Haili, Some Algebraic Properties of b-Parts of Real Numbers, Siauliai Math.Semin., vol.3, no.11 (2008), 115-121.
6
[7] M.H. Hooshmand and S. Rahimian, A study of regular grouplikes, J. Math. Ext., vol. 7, no. 4 (2013), 1{9.
7
[8] R. P. Hunter, On the structure of homogroups with applications to the theory of compact connected semigroups, Fund. Math. vol. 52 (1963), 69-102.
8
[9] A. Nagy, Special Classes of Semigroups, Kluwer Academic Publishers, 2001.
9
[10] G. Thierrin, Contribution a la theorie des equivalences dans les demi-groupes, Bull. Soc. Math. France, vol. 83 (1955), 103-159.
10
ORIGINAL_ARTICLE
When does the complement of the annihilating-ideal graph of a commutative ring admit a cut vertex?
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.
http://as.yazd.ac.ir/article_765_b8befa609c45c0b6b6a79bc456253b4a.pdf
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22
N-prime of $(0)$
B-prime of $(0)$
complement of the annihilating-ideal graph of a commutative ring
vertex cut and cut vertex of a connected graph
S.
VISWESWARAN
true
1
Saurashtra University, Rajkot
Saurashtra University, Rajkot
Saurashtra University, Rajkot
LEAD_AUTHOR
A.
PARMAR
true
2
Saurashtra University, Rajkot
Saurashtra University, Rajkot
Saurashtra University, Rajkot
AUTHOR
[1] G Aalipour, S. Akbari, M. Behboodi, R. Nikandish, M.J. Nikmehr, and F. Shaiveisi, The classication of
1
annihilating-ideal graph of commutative rings, Alg. Colloquium, 21, 249 (2014), doi:10.1143/S1005386714000200.
2
[2] G. Aalipour, S. Akbari, R. Nikandish, M.J. Nikmehr, and F. Shaiveisi, On the coloring of the annihilating-
3
ideal graph of a commutative ring, Discrete Math., 312 (2012), 2620-2625.
4
[3] .D.F. Anderson, M.C. Axtell, J.A. Stickles Zero-divisor graphs in commutative rings, in Commutative
5
Algebra, Noetherian and Non-Noetherian perspectives, M. Fontana, S.E. Kabbaj, B. Olberding, and I.
6
Swanson (Editors), Springer-Verlag, New York, 2011, 23-45.
7
[4] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Alg. 217 (1999),
8
[5] M.F. Atiyah and I.G. Macdonald, Introduction to commutative algebra, Addison-Wesley, Reading, Mas-
9
sachusetts, 1969.
10
[6] M.C. Axtell, N. Baeth, and J.A. Stickles, Cut vertices in zero-divisor graphs of nite commutative rings,
11
Comm. Alg., 39(6) (2011), 2179-2188.
12
[7] R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, Universitext, Springer, New york,
13
[8] I. . Beck, Coloring of commutative rings, J. Alg. 116 (1988), 208-226.
14
[9] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Alg. Appl. 10 (2011),
15
[10] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings II, J. Alg. Appl.10 (2011),
16
[11] B. Cotee, C. Ewing, M. Huhn, C.M. Plaut, and E.D. Weber, Cut-Sets in zero-divisor graphs of nite
17
commutative rings, Comm. Alg. 39(8) (2011), 2849-2861.
18
[12] R. Gilmer and W. Heinzer, The Laskerian property, power series rings and Noetherian spectra, Proc. Amer.
19
Math. Soc. 79(1) (1980), 13-16.
20
[13] W. Heinzer and J. Ohm, Locally Noetherian commutative rings, Trans. Amer. Math. Soc. 158(2) (1971),
21
[14] W. Heinzer and J. Ohm, On the Noetherian-like rings of E.G. Evans, Proc. Amer. Math. Soc. 34(1) (1972),
22
[15] M. Hadian, Unit action and geometric zero-divisor ideal graph, Comm. Alg. 40 (2012), 2920-2930.
23
[16] I. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago, 1974.
24
[17] T. Tamizh Chelvam and K. Selvakumar, On the connectivity of the annihilating-ideal graphs, Discuss.
25
Math. Gen. Alg. Appl. 35 (2015), 195-204.
26
[18] S. Visweswaran, Some results on the complement of the zero-divisor graph of a commutative ring, J. Alg.
27
Appl. 10(3) (2011), 573-595.
28
[19] S. Visweswaran, Some properties of the complement of the zero-divisor graph of a commutative ring, ISRN Alg. 2011 (2011), Article ID 591041, 24 pages.
29
[20] S. Visweswaran and Hiren D. Patel, Some results on the complement of the annihilating ideal graph of a
30
commutative ring, J. Algebra Appl. 14 (2015), doi: 10.1142/S0219498815500991, 23 pages.
31
[21] S. Visweswaran, When does the complement of the zero-divisor graph of a commutative ring admit a cut
32
vertex?, Palestine J. Math. 1(2) (2012), 138-147.
33
ORIGINAL_ARTICLE
Ultra and Involution Ideals in $BCK$-algebras
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.
http://as.yazd.ac.ir/article_784_59ea8d93f1f07746b0ae002a32a6a389.pdf
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23
36
$BCK$-algebra
(associative
commutative
positive implicative
implicative) ideal
ultra ideal
involution ideal
Simin
Saidi Goraghani
siminsaidi@yahoo.com
true
1
Farhangian University
Farhangian University
Farhangian University
LEAD_AUTHOR
R. A.
Borzooei
true
2
Shahid Beheshti University
Shahid Beheshti University
Shahid Beheshti University
AUTHOR
[1] R. A. Borzooei, J. Shohani, Fraction structures on bounded implicative BCK-algebras, Word Academy of
1
Science, Engineering and Thechnology, 49, 1084-1090 (2009).
2
[2] O. Heubo-Kwegna and J. B. Nganou, A Global Local Principle for BCK-modules, International Journal of
3
Algebra, 5(14), 691-702 (2011).
4
[3] Y. Huang, BCI-algebra, Science Press, Beijing (2006).
5
[4] Y. Imai and K. Iseki, On axiom systems of propositional calculi, Proceedings of the Japan Academy, 42,
6
19-21 (1966).
7
[5] K. Iseki, On ideals in BCK-algebras, Mathematics Seminar Notes, 3, 1-12 (1975).
8
[6] K. Iseki and S. Tanaka, Ideal theory of BCK-algebras, Mathematica Japonica, 21, 351-366 (1976).
9
[7] P. Jiayin, Normed BCK-algebras, Advances in Mathematics, 4, 492-500 (2011).
10
[8] J. Meng and Y. B. Jun, BCK-algebras, Kyung Moon Sa Co, seoul, (1994).
11
[9] Z. M. Samaei and M. A. N. Azadani, A Class of BCK-algebras, International Journal of Algebra, 28, 1379-1385 (2011).
12
ORIGINAL_ARTICLE
The structure of a pair of nilpotent Lie algebras
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.
http://as.yazd.ac.ir/article_785_f8abf078bb44933f3c1b0a1d39b66275.pdf
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37
47
Nilpotent Lie algebra
Pair of Lie algebras
Schur multiplier
Homayoon
Arabyani
arabyani_h@yahoo.com
true
1
Islamic Azad University
Islamic Azad University
Islamic Azad University
LEAD_AUTHOR
Hadi Hosseini
Fadravi
true
2
Islamic Azad University
Islamic Azad University
Islamic Azad University
AUTHOR
[1] J. M. Ancochea-Bermdez and M. Goze, Classication des algbres de Lie nilpotentes de dimension 7, Arch.
1
Math., 52(2) (1989), 157-185.
2
[2] H. Arabyani, F. Saeedi, M. R. R. Moghaddam and E. Khamseh, On characterizing a pair of nilpotent Lie
3
algebras by their Schur multipliers, Comm. Alg, 42 (2014), 5474-5483.
4
[3] P. Batten, Multipliers and covers of Lie algebras, PhD thesis, North Carolina State university, (1993).
5
[4] P. Batten, K. Moneyhum and E. Stitzinger, On characterizing nilpo- tent Lie algebras by their multipliers,
6
Comm. Algebra, 24(14) (1996), 4319-4330.
7
[5] P. Batten, E. Stitzinger, On covers of Lie algebras, Comm. Algebra, 24(14), (1996), 4301-4317.
8
[6] L. R. Bosko, On Schur multipliers of Lie algebras and groups of maximal class,internat. J. Algebra comput,
9
20(6), (2010), 807-821.
10
[7] S. Cicalo, W. A. de Graaf and C. Schneider, Six- dimensional nilpotent Lie algebras, Linear Algebra Appl.,
11
436(1) (2012), 163-189.
12
[8] J. Dixmier, Sur les reprsentations unitaires des groupes de Lie nilpo- tents III, Canad. J. Math., 10 (1958),
13
[9] G. Ellis, A non-abelian tensor square of Lie algebras, Glasgow Math. J. (39) (1991), 101{120.
14
[10] K. Erdmann and M. Wildon, Introduction to Lie Algebras, Springer undergraduate Mathematics series,
15
[11] P. Hardy, On characterizing nilpotent Lie algebras by their multipliers (III), Comm. Algebra. 33 (2005),
16
4205-4210.
17
[12] P. Hardy and E. Stitzinger, On characterizing nilpotent Lie algebras by their multipliers, t(L) = 3; 4; 5; 6,
18
Comm. Algebra. 26(11) (1998), 3527-3539.
19
[13] M. R. R. Moghaddam, A. R. Salemkar and K. Chiti, Some properties on the Schur multiplier of a pair of
20
groups, J. Algebra 312 (2007), 1-8.
21
[14] M. R. R. Moghaddam, A. R. Salemkar and T. Karimi, Some inequalities for the order of Schur multiplier
22
of a pair of groups, Comm. Algebra 36 (2008), 1-6.
23
[15] K. Moneyhun, Isoclinism in Lie algebras, Algebra Groups Geom., 11 (1994), 9-22.
24
[16] V. V. Morozov, Classication des algebras de Lie nilpotents de dimension 6. Izv. Vyssh. Ucheb. Zar, 4
25
(1958), 161{171.
26
[17] P. Niroomand, On dimension of the Schur multiplier of nilpotent Lie algebras,cent. Eur. J. Math., 9(1)
27
(2011), 57-64.
28
[18] P. Niroomand and F.G. Russo, A note on the Schur multiplier of a nilpotent Lie algebra, Comm. Algebra,
29
39 (2011), 1293-1297.
30
[19] M.R. Rismanchian and M. Araskhan, Some inequalities for the dimension of the Schur multilier of a pair
31
of (nilpotent) Lie algebras, J. Algebra , 352 (2012), 173-179.
32
[20] M. R. Rismanchian and M. Araskhan, Some properties on the Schur multiplier of a pair of Lie algebras,
33
J. Algebra Appl., 11 (2012), 1250011(9 pages).
34
[21] D. J. S. Robinson, A Course in the Theory of Groups, Springer Verlag, New York, 1982.
35
[22] F. Saeedi, A. R. Salemkar and B. Edalatzadeh, The commutator subalgebra and Schur multiplier of a pair
36
of nilpotent Lie algebras, J. Lie Theory, 21 (2011), 491-498.
37
[23] F. Saeedi, H. Arabyani and P. Niroomand, On dimension of the Schur multiplier of nilpotent Lie algebras
38
(II),(submitted).
39
[24] A. R. Salemkar, V. Alamian, H. Mohammadzadeh, Some properties of the Schur multiplier and covers of
40
Lie algebras, Comm. Algebra. 36(2) (2008), 697-707.
41
[25] A. R. Salemkar and S. Alizadeh Niri, Bounds for the dimension of the Schur multiplier of a pair of nilpotent Lie algebras, Asian-Eur. J. Math. 5 (2012), 1250059(9 pages).
42
[26] B. Yankosky, On the multiplier of a Lie algebra, J. Lie Theory, 13(1) (2003), 1-6.
43
ORIGINAL_ARTICLE
On the nil-clean matrix over a UFD
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.
http://as.yazd.ac.ir/article_803_7a98829c79d5ccc6521ac399e996e7bb.pdf
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55
Rank of a matrix
Idempotent matrix
Nilpotent matrix
Nil-clean matrix
Strongly nil-clean matrix
Somayeh
Hadjirezaei
s.hajirezaei@vru.ac.ir
true
1
Vali-e-Asr University of Rafsanjan
Vali-e-Asr University of Rafsanjan
Vali-e-Asr University of Rafsanjan
LEAD_AUTHOR
Somayeh
Karimzadeh
true
2
Vali-e-Asr University of Rafsanjan
Vali-e-Asr University of Rafsanjan
Vali-e-Asr University of Rafsanjan
AUTHOR
[1] S. Breaz, G. Calugaranu, P. Danchev, T. Micu, Nil-clean matrix rings, Linear Algebra Appl., vol. 439, no. 1 (2013), 3115-3119.
1
[2] A. J. Diesl, Classes of strongly clean rings, Phd thesis, University of California, Berkeley, 2006.
2
[3] A. J. Diesl, Nil clean rings, J. Algebra, vol. 383, (2013), 197-211.
3
[4] T. W. Hungerford, Algebra, Springer-Verlag, 1980.
4
[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), 7-12.
5
[6] W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc., vol. 229 (1977), 269-278.
6
[7] G. Song, X. Guo, Diagonability of idempotent matrices over noncommutative rings, Linear Algebra Appl., vol. 297, no. 1 (1999), 1-7.
7
ORIGINAL_ARTICLE
$z^\circ$-filters and related ideals in $C(X)$
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.
http://as.yazd.ac.ir/article_807_bb25ddc73dfd82df981f87a48bcc5e25.pdf
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57
66
$z^circ$-filter
prime $z^circ$-filter
ci-free $z^circ$-filter
i-free $z^circ$-filter
$z^circ$-ultrafilter
i-compact
Rostam
Mohamadian
mohamadian_r@scu.ac.ir
true
1
Shahid Chamran University of Ahvaz
Shahid Chamran University of Ahvaz
Shahid Chamran University of Ahvaz
LEAD_AUTHOR
[1] F. Azarpanah, On almost P-spaces, Far East J. Math. Sci., Special Volume, 121{132 (2000).
1
[2] F. Azarpanah, O.A.S. Karamzadeh and A. Rezaei Aliabad, On z-ideals in C(X), Fund. Math., 160, 15{25 (1999).
2
[3] F. Azarpanah, O.A.S. Karamzadeh and A. Rezaei Aliabad, On ideal consisting entirely of zerodivisor, Comm. Algebra, 28(2), 1061{1073 (2000).
3
[4] F. Azarpanah and M. Karavan, On nonregular ideals and z-ideal in C(X), Cech. Math. J., 55(130), 397{407 (2005).
4
[5] F. Azarpanah and R. Mohamadian, pz-ideals and pz-ideals in C(X), Acta. Math. Sinica., English Series, 23,
5
989{996 (2007).
6
[6] R. Engelking, General Topology, PWN-Polish Sci Publ, 1977.
7
[7] L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag, New York, 1976.
8
[8] G. Mason, z-ideals and prime ideals, J. Algebra, 26: 280{297 (1973).
9