THE ANNIHILATING-IDEAL GRAPH OF A RING
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Let S be a semigroup with 0 and R be a ring with 1. We ex- tend the definition of the zero-divisor graphs of commutative semigroups to not necessarily commutative semigroups. We define an annihilating- ideal graph of a ring as a special type of zero-divisor graph of a semigroup. We introduce two ways to define the zero-divisor graphs of semigroups. The first definition gives a directed graph Γ(S), and the other definition yields an undirected graph Γ―(S). It is shown that Γ(S) is not necessar- ily connected, but Γ―(S) is always connected and diam(Γ―(S)) 3. For a ring R define a directed graph APOG(R) to be equal to Γ(IPO(R)), where IPO(R) is a semigroup consisting of all products of two one-sided ideals of R, and define an undirected graph APOG―(R) to be equal to Γ―(IPO(R)). We show that R is an Artinian (resp., Noetherian) ring if and only if APOG(R) has DCC (resp., ACC) on some special subset of its vertices. Also, it is shown that APOG―(R) is a complete graph if and only if either (D(R))2 = 0, R is a direct product of two division rings, or R is a local ring with maximal ideal m such that IPO(R) = {0,m,m2,R}. Finally, we investigate the diameter and the girth of square matrix rings over commutative rings Mn×n(R) where n ≥ 2.
Abstract
Let S be a semigroup with 0 and R be a ring with 1. We ex- tend the definition of the zero-divisor graphs of commutative semigroups to not necessarily commutative semigroups. We define an annihilating- ideal graph of a ring as a special type of zero-divisor graph of a semigroup. We introduce two ways to define the zero-divisor graphs of semigroups. The first definition gives a directed graph Γ(S), and the other definition yields an undirected graph Γ―(S). It is shown that Γ(S) is not necessar- ily connected, but Γ―(S) is always connected and diam(Γ―(S)) 3. For a ring R define a directed graph APOG(R) to be equal to Γ(IPO(R)), where IPO(R) is a semigroup consisting of all products of two one-sided ideals of R, and define an undirected graph APOG―(R) to be equal to Γ―(IPO(R)). We show that R is an Artinian (resp., Noetherian) ring if and only if APOG(R) has DCC (resp., ACC) on some special subset of its vertices. Also, it is shown that APOG―(R) is a complete graph if and only if either (D(R))2 = 0, R is a direct product of two division rings, or R is a local ring with maximal ideal m such that IPO(R) = {0,m,m2,R}. Finally, we investigate the diameter and the girth of square matrix rings over commutative rings Mn×n(R) where n ≥ 2.
참고문헌 (25)
- G. Aalipour, "The classification of the annihilating-ideal graphs of commutative rings", Algebra Colloq, 21(2), pp.249~256 (2014)
- G. Aalipour, "On the coloring of the annihilating-ideal graph of a commutative ring", Discrete Math, 312(17), pp.2620~2626 (2012)
- G. Aalipour, "Minimal prime ideals and cycles in annihilating-ideal graphs", Rocky Mountain J. Math, 43(5), pp.1415~1425 (2013)
- S. Akbari, "On the zero-divisor graph of a commutative ring", J. Algebra, 274(2), pp.847~855 (2004)
- S. Akbari, "Zero-divisor graphs of non-commutative rings", J. Algebra, 269(2), pp.462~479 (2006)
- S. Akbari, "On zero-divisor graphs of finite rings", J. Algebra, 314(1), pp.168~184 (2007)
- F. Aliniaeifard, "Rings whose annihilating-ideal graphs have positive genus", J. Algebra Appl, 11(3), pp.13~ (2012)
- F. Aliniaeifard, "Commutative rings whose zero-divisor graphs have positive genus", Comm. Al-gebra, 41(10), pp.3629~3634 (2013)
- D. F. Anderson, "Zero-divisor graphs, von Neumann regular rings, and Boolean algebras", J. Pure Appl. Algebra, 180(3), pp.221~241 (2003)
- D. F. Anderson, "The zero-divisor graph of a commutative ring", J. Algebra, 217(2), pp.434~447 (1999)
- I. Beck, "Coloring of commutative rings", J. Algebra, 116(1), pp.208~226 (1988)
- M. Behboodi, "The annihilating-ideal graph of commutative rings I", J. Algebra Appl, 10(4), pp.727~739 (2011)
- M. Behboodi, "The annihilating-ideal graph of commutative rings II", J. Algebra Appl, 10(4), pp.741~753 (2011)
- F. DeMeyer, "Zero-divisor graphs of semigroups", J. Algebra, 283(1), pp.190~198 (2005)
- F. DeMeyer, "The Zero-divisor graph of a commutative semigroup", Semigroup Forum, 65(2), pp.206~214 (2002)
- F. DeMeyer, "Automorphisms and zero-divisor graphs of commutative rings, Commutative rings", pp.25~37 (2002), Nova Sci. Publ.
- N. S. Karamzadeh, "On Artinian modules over Duo rings", Comm. Algebra, 38(9), pp.3521~3531 (2010)
- T. Y. Lam, "A First Course in Non-Commutative Rings" (1991), Springer-Verlag
- S. B. Mulay, "Cycles and symmetries of zero-divisors", Comm. Algebra, 30(7), pp.3533~3558 (2002)
- S. Redmond, "The zero-divisor graph of a non-commutative ring, Commutative rings", pp.39~47 (2002), Nova Sci. Publ.
- S. Redmond, "Structure in the zero-divisor graph of a noncommutative ring", Houston J. Math, 30(2), pp.345~355 (2004)
- T. S. Wu, "On directed zero-divisor graphs of finite rings", Discrete Math, 296(1), pp.73~86 (2005)
- T. S. Wu, "Zero-divisor semigroups and refinements of a star graph", Discrete Math, 309(8), pp.2510~2518 (2009)
- T. S. Wu, "Zero-divisor semigroups and some simple graphs", Comm. Al-gebra, 34(8), pp.3043~3052 (2006)
- T. S. Wu, "Sub-semigroups determined by the zero-divisor graph", Discrete Math, 308(22), pp.5122~5135 (2008)