ORIGINAL_ARTICLE
Birkhoff's Theorem from a geometric perspective: A simple example
From Hilbert's theorem of zeroes, and from Noether's ideal theory, Birkhoff derived certain algebraic concepts (as explained by Tholen) that have a dual significance in general toposes, similar to their role in the original examples of algebraic geometry. I will describe a simple example that illustrates some of the aspects of this relationship. The dualization from algebra to geometry in the basic Grothendieck spirit can be accomplished (without intervention of topological spaces) by the following method, known as Isbell conjugacy.
http://cgasa.sbu.ac.ir/article_12425_b4ce2ab0ae3a843f00ff011b054f918b.pdf
2016-02-01T11:23:20
2018-04-20T11:23:20
1
8
Grothendieck spectrum
Cantor
Boole
Hilbert
Birkhoff: Existence and Sufficiency of generalized points
Reflexive Graphs
F. William
Lawvere
wlawvere@buffalo.edu
true
1
Department of Mathematics, University at Buffalo, Buffalo, New York 14260-2900, United States of America.
Department of Mathematics, University at Buffalo, Buffalo, New York 14260-2900, United States of America.
Department of Mathematics, University at Buffalo, Buffalo, New York 14260-2900, United States of America.
LEAD_AUTHOR
[1] G. Birkhoff, Subdirect unions in universal algebra, Bull. Amer. Math. Soc. 50 (1944),
1
[2] G. Cantor, Beiträge zur Begründung der transfiniten Mengenlehre, Math. Ann. 46
2
(1895) 481-512.
3
[3] J.R. Isbell, Small subcategories and completeness, Math. Syst. Theory 2 (1968),
4
[4] F.W. Lawvere, Unity and identity of opposites in calculus and Physics, Appl. Categ.
5
Structures, 4 (1996), 167-174.
6
[5] F.W. Lawvere, Taking categories seriously, Reprints in Theory Appl. Categ. 8
7
(2005), 1-24.
8
[6] F.W. Lawvere, Axiomatic cohesion, Theory Appl. Categ. 19 (2007), 41-49.
9
[7] F.W. Lawvere, Core varieties, extensivity, and rig geometry, Theory Appl. Categ.
10
20 (2008), 497-503.
11
[8] F.W. Lawvere and M. Menni, Internal choice holds in the discrete part of any co-
12
hesive topos satisfying stable connected codiscreteness, Theory Appl. Categ. 30(26)
13
(2015), 909-932.
14
[9] F.W. Lawvere and S.H. Schanuel, “Conceptual Mathematics”, Cambridge Univer-
15
sity Press, 2nd edition, 2009.
16
[10] W. Tholen, Nullstellen and subdirect representation, Appl. Categ. Structures 22
17
(2014), 907-929.
18
ORIGINAL_ARTICLE
Steps toward the weak higher category of weak higher categories in the globular setting
We start this article by rebuilding higher operads of weak higher transformations, and correct those in \cite{Cambat}. As in \cite{Cambat} we propose an operadic approach for weak higher $n$-transformations, for each $n\in\mathbb{N}$, where such weak higher $n$-transformations are seen as algebras for specific contractible higher operads. The last chapter of this article asserts that, up to precise hypotheses, the higher operad $B^{0}_{C}$ of Batanin and the terminal higher operad $B^{0}_{S_{u}}$, both have the fractal property. In other words we isolate the precise technical difficulties behind a major problem in globular higher category theory, namely, that of proving the existence of the globular weak higher category of globular weak higher categories.
http://cgasa.sbu.ac.ir/article_11180_b13cacfd9afe5780932141c269d0add6.pdf
2016-02-01T11:23:20
2018-04-20T11:23:20
9
42
globular sets
weak higher categories
weak higher transformations
higher operads
Camell
Kachour
camell.kachour@gmail.com
true
1
Department of Mathematics, Macquarie University, North Ryde, NSW 2109, Australia.
Department of Mathematics, Macquarie University, North Ryde, NSW 2109, Australia.
Department of Mathematics, Macquarie University, North Ryde, NSW 2109, Australia.
LEAD_AUTHOR
[1] J. Adámek and J. Rosick´ y, “Locally Presentable and Accessible Categories”, Cam-
1
bridge University Press, 1994.
2
[2] M. Batanin, Monoidal globular categories as a natural environment for the theory
3
of weak-n-categories, Adv. Math. 136 (1998), 39–103.
4
[3] F. Borceux, “Handbook of Categorical Algebra, Vol. 2”, Cambridge University
5
Press, 1994.
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[4] L. Coppey and Ch. Lair, “Le¸ cons de th´ eorie des esquisses”, Universit´ e Paris VII,
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[5] C. Hermida, Representable multicategories, Adv. Math. 151(2) (2000), 164–225.
8
[6] K. Kachour, D´ efinition alg´ ebrique des cellules non-strictes, Cah. Topol. G´ eom.
9
Diff´ er. Cat´ eg. 1 (2008), 1–68.
10
[7] C. Kachour, Operadic definition of the non-strict cells, Cah. Topol. G´ eom. Diff´ er.
11
Cat´ eg. 4 (2011), 1–48.
12
[8] C. Kachour, Correction to the paper “Operadic definition of the non-strict cells”
13
(2011), Cah. Topol. G´ eom. Diff´ er. Cat´ eg. 54 (2013), 75-80.
14
[9] C. Kachour, “Aspects of Globular Higher Category Theory”, Ph.D. Thesis, Mac-
15
quarie University, 2013.
16
[10] C. Kachour, ω-Operads of coendomorphisms and fractal ω-operads for higher struc-
17
tures, Categ. General Alg. Structures Appl. 3(1) (2015), 65–88.
18
[11] C. Kachour, Operads of higher transformations for globular sets and for higher
19
magmas, Categ. General Alg. Structures Appl. 3(1) (2015), 89–111.
20
[12] C. Kachour and J. Penon, Batanin ω-Operads for the weak higher transformations
21
and stable pseudo-algebras, Work in progress (2015).
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[13] G.M. Kelly, A unified treatment of transfinite constructions for free algebras, free
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monoids, colimits, associated sheaves, and so on, Bull. Aust. Math. Soc. 22 (1980),
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1–83.
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[14] T. Leinster, “Higher Operads, Higher Categories”, London Math. Soc. Lect. Note
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Series, Cambridge University Press 298, 2004.
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[15] M. Makkai and R. Par´ e, “Accessible Categories: The Foundations of Categorical
28
Model Theory”, American Mathematical Society, 1989.
29
ORIGINAL_ARTICLE
Basic notions and properties of ordered semihyperrings
In this paper, we introduce the concept of semihyperring $(R,+,\cdot)$ together with a suitable partial order $\le$. Moreover, we introduce and study hyperideals in ordered semihyperrings. Simple ordered semihyperrings are defined and its characterizations are obtained. Finally, we study some properties of quasi-simple and $B$-simple ordered semihyperrings.
http://cgasa.sbu.ac.ir/article_11181_a73c9c7bbdb038f75ed62901bc042c2a.pdf
2016-02-01T11:23:20
2018-04-20T11:23:20
43
62
ordered semihyperring
hyperideal
simple
quasi-simple
$B$-simple
B.
Davvaz
davvaz@yazd.ac.ir
true
1
Department of
Mathematics, Yazd University, Yazd, Iran.
Department of
Mathematics, Yazd University, Yazd, Iran.
Department of
Mathematics, Yazd University, Yazd, Iran.
LEAD_AUTHOR
S.
Omidi
omidi.saber@yahoo.com
true
2
Department of
Mathematics, Yazd University, Yazd, Iran.
Department of
Mathematics, Yazd University, Yazd, Iran.
Department of
Mathematics, Yazd University, Yazd, Iran.
AUTHOR
[1] N.G. Alimov, On ordered semigroups, Izvestiya Akad. Nauk SSSR. 14 (1950), 569-
1
[2] R. Ameri and H. Hedayati, On k-hyperideals of semihyperrings, J. Discrete Math.
2
Sci. Cryptogr. 10(1) (2007), 41-54.
3
[3] A. Asokkumar, Class of semihyperrings from partitions of a set, Ratio Math. 25
4
(2013), 3-14.
5
[4] M. Bakhshi and R. A. Borzooei, Ordered polygroups, Ratio Math. 24 (2013), 31-40.
6
[5] T. Changphas and B. Davvaz, Properties of hyperideals in ordered semihypergroups,
7
Ital. J. Pure Appl. Math. 33 (2014), 425-432.
8
[6] J. Chvalina, “Commutative hypergroups in the sence of Marty and ordered sets”
9
Proceedings of the Summer School in General Algebra and Ordered Sets, Olomouck,
10
1994, 19-30.
11
[7] J. Chvalina and J. Moucka, Hypergroups determined by orderings with regular endo-
12
morphism monoids, Ital. J. Pure Appl. Math. 16 (2004), 227-242.
13
[8] A.H. Clifford, Totally ordered commutative semigroups, Bull. Amer. Math. Soc. 64
14
(1958), 305-316.
15
[9] P. Corsini, “Prolegomena of Hypergroup Theory”, Second edition, Aviani Editore,
16
Italy, 1993.
17
[10] P. Corsini and V. Leoreanu, “Applications of Hyperstructure Theory” Adv. Math.,
18
Kluwer Academic Publishers, Dordrecht, 2003.
19
[11] B. Davvaz, Isomorphism theorems of hyperrings, Indian J. Pure Appl. Math. 35(3)
20
(2004), 321-331.
21
[12] B. Davvaz, Rings derived from semihyperrings, Algebras Groups Geom. 20 (2003),
22
[13] B. Davvaz, Some results on congruences in semihypergroups, Bull. Malays. Math.
23
Sci. Soc. 23(2) (2000), 53-58.
24
[14] B. Davvaz, “Polygroup Theory and Related Systems”, World scientific publishing
25
Co. Pte. Ltd., Hackensack, NJ, 2013.
26
[15] B. Davvaz and V. Leoreanu-Fotea, “Hyperring Theory and Applications”, Interna-
27
tional Academic Press, Palm Harbor, USA, 2007.
28
[16] B. Davvaz, P. Corsini and T. Changphas, Relationship between ordered semihy-
29
pergroups and ordered semigroups by using pseudoorder, European J. Combin. 44
30
(2015), 208-217.
31
[17] L. Fuchs, “Partially Ordered Algebraic Systems”, Pergamon Press, New York, 1963.
32
[18] A.P. Gan and Y.L. Jiang, On ordered ideals in ordered semirings, J. Math. Res.
33
Exposition 31(6) (2011), 989-996.
34
[19] L. Gillman and M. Jerison, “Rings of Continuous Functions”, Van Nostrand Com-
35
pany, Inc., Princeton, 1960.
36
[20] J.S. Golan, “Semirings and Their Applications”, Kluwer Academic Publishers, Dor-
37
drecht, 1999.
38
[21] R.A. Good and D.R. Hughes, Associated groups for a semigroup, Bull. Amer. Math.
39
Soc. 58 (1952), 624-625.
40
[22] D. Heidari and B. Davvaz, On ordered hyperstructures, Politehn. Univ. Bucharest
41
Sci. Bull. Ser. A Appl. Math. Phys. 73(2) (2011), 85-96.
42
[23] S. Hoskova, Representation of quasi-order hypergroups, Glob. J. Pure Appl. Math.
43
1 (2005), 173-176.
44
[24] S. Hoskova, Upper order hypergroups as a reflective subcategory of subquasiorder
45
hypergroups, Ital. J. Pure Appl. Math. 20 (2006), 215-222.
46
[25] X. Huang, Y. Yin and J. Zhan, Characterizations of semihyperrings by their (? γ ,? γ ∨
47
q δ )-fuzzy hyperideals, J. Appl. Math. 2013 (2013), 13 pages.
48
[26] K. Iseki, Quasi-ideals in semirings without zero, Proc. Japan Acad. 34 (1958), 79-84.
49
[27] N. Kehayopulu, Note on bi-ideals in ordered semigroups, Pure Math. Appl. 6 (1955),
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[28] N. Kehayopulu and M. Tsingelis, On subdirectly irreducible ordered semigroups,
51
Semigroup Forum 50 (1995), 161-177.
52
[29] N. Kehayopulu and M. Tsingelis, Pseudoorder in ordered semigroups, Semigroup
53
Forum 50 (1995), 389-392.
54
[30] N. Kehayopulu, J.S. Ponizovskii and M. Tsingelis, Bi-ideals in ordered semigroups
55
and ordered groups, J. Math. Sci. 112(4) (2002), 4353-4354.
56
[31] M. Krasner, A class of hyperrings and hyperfields, Internat. J. Math. Math. Sci. 6(2)
57
(1983), 307-312.
58
[32] S. Lajos, On regular duo rings, Proc. Japan Acad. 45 (1969), 157-158.
59
[33] F. Marty, Sur une generalisation de la notion de groupe, 8 iem Congress Math. Scan-
60
dinaves, Stockholm (1934), 45-49.
61
[34] J. Mittas, Hypergroupes canoniques, Math. Balkanica, 2 (1972), 165-179.
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[35] H. J. L. Roux, A note on prime and semiprime bi-ideals, Kyungpook Math. J. 35
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(1995), 243-247.
64
[36] T. Saito, Regular elements in an ordered semigroup, Pacific J. Math. 13 (1963),
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[37] S. Spartalis, A class of hyperrings, Rivista Mat. Pura Appl. 4 (1989), 56-64.
66
[38] O. Steinfeld, “Quasi-ideals in rings and semigroups”, Akademiai Kiado, Budapest,
67
[39] D. Stratigopoulos, Hyperanneaux, hypercorps, hypermodules, hyperspaces vectoriels
68
et leurs proprietes elementaires, C. R. Acad. Sci., Paris A (269) (1969), 489-492.
69
[40] H. S. Vandiver, Note on a simple type of algebra in which cancellation law of addition
70
does not hold, Bull. Amer. Math. Soc. 40 (1934), 914-920.
71
[41] T. Vougiouklis, On some representations of hypergroups, Ann. Sci. Univ. Clermont-
72
Ferrand II Math. 26 (1990), 21-29.
73
[42] T. Vougiouklis, “Hyperstructures and Their Representations”, Hadronic Press Inc.,
74
Florida, 1994.
75
[43] X.Z.Xu and J.Y.Ma, A note on minimal bi-ideals in ordered semigroups, Southeast
76
Asian Bull. Math. 27 (2003), 149-154.
77
ORIGINAL_ARTICLE
A characterization of finitely generated multiplication modules
Let $R$ be a commutative ring with identity and $M$ be a finitely generated unital $R$-module. In this paper, first we give necessary and sufficient conditions that a finitely generated module to be a multiplication module. Moreover, we investigate some conditions which imply that the module $M$ is the direct sum of some cyclic modules and free modules. Then some properties of Fitting ideals of modules which are the direct sum of finitely generated module and finitely generated multiplication module are shown. Finally, we study some properties of modules that are the direct sum of multiplication modules in terms of Fitting ideals.
http://cgasa.sbu.ac.ir/article_12667_7069a62adca415a4a7178c2d5b4804a7.pdf
2016-02-01T11:23:20
2018-04-20T11:23:20
63
74
Fitting ideals
multiplication module
projective module
Somayeh
Karimzadeh
karimzadeh@vru.ac.ir
true
1
Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran.
Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran.
Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran.
LEAD_AUTHOR
Somayeh
Hadjirezaei
s.hajirezaei@vru.ac.ir
true
2
Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran.
Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran.
Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran.
AUTHOR
[1] I. Akharraz, M. E. Charkani, Induced modules by endomorphism of nitely generated
1
modules, Int. J. Algebra 3(12) (2009), 589-597.
2
[2] A. Barnard, Multiplication modules, J. Algebra 71 (1981), 174-178.
3
[3] W. C. Brown, Matrices Over Commutative Rings", Pure Appl. Math. 169, Marcel
4
Dekker Inc., New York, 1993.
5
[4] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry",
6
Springer-Verlag, New York, 1995.
7
[5] Z. A. El-Bast, P.F. Smith, Multiplication modules, Comm. Algebra 16 (1988),
8
[6] N. S. Gopalakrishnan, Commutative Algebra", Oxonian Press, New Delhi, 1984.
9
[7] J. Ohm, On the rst nonzero Fitting ideal of a module, J. Algebra 320 (2008),
10
[8] J. Rotman, An Introduction to Homological Algebra", Springer, 2008.
11
[9] Y. Tiras, M. Alkan, Prime modules and submodules, Comm. Algebra, 31 (2003),
12
5253-5261.
13
[10] P. Vamos, Finitely generated Artinian and distributive modules are cyclic, Bull.
14
London Math. Soc. 10 (1978), 287-288.
15
ORIGINAL_ARTICLE
The ring of real-continuous functions on a topoframe
A topoframe, denoted by $L_{ \tau}$, is a pair $(L, \tau)$ consisting of a frame $L$ and a subframe $ \tau $ all of whose elements are complementary elements in $L$. In this paper, we define and study the notions of a $\tau $-real-continuous function on a frame $L$ and the set of real continuous functions $\mathcal{R}L_\tau $ as an $f$-ring. We show that $\mathcal{R}L_{ \tau}$ is actually a generalization of the ring $C(X)$ of all real-valued continuous functions on a completely regular Hausdorff space $X$. In addition, we show that $\mathcal{R}L_{ \tau}$ is isomorphic to a sub-$f$-ring of $\mathcal{R}\tau .$ Let ${\tau}$ be a topoframe on a frame $L$. The frame map $\alpha\in\mathcal{R}\tau $ is called $L$-{\it extendable} real continuous function if and only if for every $r\in \mathbb{R}$, $\bigvee^{L}_{r\in \mathbb R} (\alpha(-,r)\vee\alpha(r,-))'=\top.$ Finally, we prove that $\mathcal{R}^{L}{\tau}\cong \mathcal{R}L_{\tau}$ as $f$-rings, where $\mathcal{R}^{L}{\tau}$ is the set all of $L$-extendable real continuous functions of $ \mathcal{R}\tau $.
http://cgasa.sbu.ac.ir/article_13184_2f80da4a155068ca432d536a7217a6ab.pdf
2016-02-01T11:23:20
2018-04-20T11:23:20
75
94
Frame
Topoframe
Ring of real continuous
functions
Archimedean ring
$f$-ring
Ali Akbar
Estaji
aaestaji@hsu.ac.ir
true
1
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
AUTHOR
Abolghasem
Karimi Feizabadi
akarimi@gorganiau.ac.ir
true
2
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
AUTHOR
Mohammad
Zarghani
zarghanim@yahoo.com
true
3
Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
AUTHOR
[1] R.N. Ball and J. Walters-Wayland, C- and C-quotients in pointfree topology, Disser-
1
tationes Math. (Rozprawy Mat.) 412 (2002), 1-61.
2
[2] R.N. Ball and A.W. Hager, On the localic Yosida representation of an archimedean
3
lattice ordered group with weak unit, J. Pure Appl. Algebra 70 (1991), 17-43.
4
[3] B. Banaschewski, On the function ring functor in pointfree topology, Appl. Categ.
5
Structures 13 (2005), 305-328.
6
[4] B. Banaschewski, The real numbers in pointfree topology", Textos de Mathematica
7
(Series B) 12, University of Coimbra, 1997.
8
[5] T. Dube, A note on the socle of certain types of f-rings, Bull. Iranian Math. Soc.
9
38(2) (2012), 517-528.
10
[6] T. Dube, Extending and contracting maximal ideals in the function rings of pointfree
11
topology, Bull. Math. Soc. Sci. Math. Roumanie 55(103)(4) (2012), 365-374.
12
[7] T. Dube, Some algebraic characterizations of F-frames, Algebra Universalis 62 (2009),
13
[8] T. Dube, Some ring-theoretic properties of almost P-frames, Algebra Universalis 60
14
(2009), 145-162.
15
[9] M.M. Ebrahimi and A. Karimi Feizabadi, Prime representation of real Riesz maps,
16
Algebra Universalis 54 (2005), 291-299.
17
[10] A.A. Estaji, A. Karimi Feizabadi, and M. Abedi, Strongly xed ideals in C(L) and
18
compact frames, Archivum Mathematicum(Brno), Tomus 51 (2015), 1-12.
19
[11] A.A. Estaji, A. Karimi Feizabadi, and M. Abedi, Zero set in pointfree topology and
20
strongly z-ideals, Bull. Iranian Math. Soc. 41(5) (2015), 1071-1084.
21
[12] M. J. Ferreira, J. Gutierrez Garca, J. Picado, Completely normal frames and real-
22
valued functions, Topology Appl. 156 (2009), 2932-2941.
23
[13] L. Gillman and M. Jerison, Rings of continuous functions", Springer-Verlag, 1976.
24
[14] C.R.A. Gilmour, Realcompact spaces and regular -frames, Math. Proc. Camb. Phil.
25
Soc. 96 (1984) 73-79.
26
[15] A. Karimi Feizabadi, A.A. Estaji, and M. Zarghani, The ring of real-valued functions
27
on a frame, Preprint.
28
[16] P.T. Johnstone, Stone Spaces", Cambridge University Press, Cambridge, 1982.
29
[17] J. Picado and A. Pultr, Frames and Locales: topology without points", Frontiers
30
in Mathematics, Springer, Basel 2012.
31
[18] M. Zarghani, A.A. Estaji, and A. Karimi Feizabadi, Modied pointfree topology,
32
ORIGINAL_ARTICLE
On zero divisor graph of unique product monoid rings over Noetherian reversible ring
Let $R$ be an associative ring with identity and $Z^*(R)$ be its set of non-zero zero divisors. The zero-divisor graph of $R$, denoted by $\Gamma(R)$, is the graph whose vertices are the non-zero zero-divisors of $R$, and two distinct vertices $r$ and $s$ are adjacent if and only if $rs=0$ or $sr=0$. In this paper, we bring some results about undirected zero-divisor graph of a monoid ring over reversible right (or left) Noetherian ring $R$. We essentially classify the diameter-structure of this graph and show that $0\leq \mbox{diam}(\Gamma(R))\leq \mbox{diam}(\Gamma(R[M]))\leq 3$. Moreover, we give a characterization for the possible diam$(\Gamma(R))$ and diam$(\Gamma(R[M]))$, when $R$ is a reversible Noetherian ring and $M$ is a u.p.-monoid. Also, we study relations between the girth of $\Gamma(R)$ and that of $\Gamma(R[M])$.
http://cgasa.sbu.ac.ir/article_13185_9afc1a95b9340cdc8d14a1cee3b2fe5c.pdf
2016-02-01T11:23:20
2018-04-20T11:23:20
95
114
Zero-divisor graphs
diameter
Girth
Reversible rings
Polynomial rings
Unique product monoids
Monoid rings
Ebrahim
Hashemi
eb_hashemi@shahroodut.ac.i
true
1
Department of Mathematics, Shahrood University of Technology, Shahrood, Iran, P.O. Box: 316-3619995161.
Department of Mathematics, Shahrood University of Technology, Shahrood, Iran, P.O. Box: 316-3619995161.
Department of Mathematics, Shahrood University of Technology, Shahrood, Iran, P.O. Box: 316-3619995161.
LEAD_AUTHOR
Abdollah
Alhevaz
a.alhevaz@shahroodut.ac.ir
true
2
Department of Mathematics, Shahrood University of Technology, Shahrood, Iran, P.O. Box: 316-3619995161.
Department of Mathematics, Shahrood University of Technology, Shahrood, Iran, P.O. Box: 316-3619995161.
Department of Mathematics, Shahrood University of Technology, Shahrood, Iran, P.O. Box: 316-3619995161.
AUTHOR
Eshag
Yoonesian
eshagh.eshag@gmail.com
true
3
Department of Mathematics, Shahrood University of Technology, Shahrood, Iran, P.O. Box: 316-3619995161.
Department of Mathematics, Shahrood University of Technology, Shahrood, Iran, P.O. Box: 316-3619995161.
Department of Mathematics, Shahrood University of Technology, Shahrood, Iran, P.O. Box: 316-3619995161.
AUTHOR
[1] S. Akbari and A. Mohammadian, Zero-divisor graphs of non-commutative rings, J.
1
Algebra 296 (2006), 462-479.
2
[2] A. Alhevaz and D. Kiani, On zero divisors in skew inverse Laurent series over
3
noncommutative rings, Comm. Algebra 42(2) (2014), 469-487.
4
[3] D.D. Anderson and V. Camillo, Semigroups and rings whose zero products commute,
5
Comm. Algebra 27(6) (1999), 2847-2852.
6
[4] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring,
7
J. Algebra 217 (1999), 434-447.
8
[5] D.F. Anderson and S.B. Mulay, On the diameter and girth of a zero-divisor graph,
9
J. Pure Appl. Algebra 210 (2007), 543-550.
10
[6] D.D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra
11
159 (1993), 500-514.
12
[7] M. Axtell, J. Coykendall, and J. Stickles, Zero-divisor graphs of polynomials and
13
power series over commutative rings, Comm. Algebra 33 (2005), 2043-2050.
14
[8] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), 208-226.
15
[9] G.F. Birkenmeier and J.K. Park, Triangular matrix representations of ring exten-
16
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