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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.
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참고문헌 (25)
참고문헌 신청
D. F. Anderson / 1999 / The zero-divisor graph of a commutative ring / J. Algebra 217 (2) : 434 ~ 447
D. F. Anderson / 2003 / Zero-divisor graphs, von Neumann regular rings, and Boolean algebras / J. Pure Appl. Algebra 180 (3) : 221 ~ 241
F. Aliniaeifard / 2012 / Rings whose annihilating-ideal graphs have positive genus / J. Algebra Appl 11 (3) : 13 ~
F. Aliniaeifard / 2013 / Commutative rings whose zero-divisor graphs have positive genus / Comm. Al-gebra 41 (10) : 3629 ~ 3634
F. DeMeyer / 2002 / Automorphisms and zero-divisor graphs of commutative rings, Commutative rings / Nova Sci. Publ. : 25 ~ 37
D. F. Anderson / 2003 / Zero-divisor graphs, von Neumann regular rings, and Boolean algebras / J. Pure Appl. Algebra 180 (3) : 221 ~ 241
F. Aliniaeifard / 2012 / Rings whose annihilating-ideal graphs have positive genus / J. Algebra Appl 11 (3) : 13 ~
F. Aliniaeifard / 2013 / Commutative rings whose zero-divisor graphs have positive genus / Comm. Al-gebra 41 (10) : 3629 ~ 3634
F. DeMeyer / 2002 / Automorphisms and zero-divisor graphs of commutative rings, Commutative rings / Nova Sci. Publ. : 25 ~ 37
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