This paper answers the question of Anderson, Frazier, Lauve, and Livingston: for which finite commutative rings R is the zero-divisor graph Γ (R) planar? We build upon and extend work of Akbari, Maimani, and Yassemi, who proved that if R is any local ring with more than 32 elements, and R is not a field, then Γ (R) is not planar. They left open the question: "Is it true that, for any local ring R of cardinality 32, which is not a field, Γ (R) is not planar?" In this paper we answer this question in the affirmative. We prove that if R is any local ring with more than 27 elements, and R is not a field, then Γ (R) is not planar. Moreover, we determine all finite commutative local rings whose zero-divisor graph is planar.

Abstract. Let R be a commutative noetherian ring and let E be the minimal injective cogenerator of the category of R-modules. A module M is said to be reflexive with respect to E if the natural evaluation map from M to Hom R (Hom R (M, E), E) is an isomorphism. We give a classification of modules which are reflexive with respect to E. A module M is reflexive with respect to E if and only if M has a finitely generated submodule S such that M/S is artinian and R/ ann(M ) is a complete semi-local ring.Matlis and Gabriel in [7] and [5] considered modules over a complete local ring R. They showed that if the dual of an R-module is taken with respect to E R (k) (the injective envelope of the residue field k of R), then finitely generated and artinian modules are reflexive.Various authors have considered related questions. For example, dropping the condition that R be complete or weakening local to semilocalIn this paper we let R be any commutative noetherian ring and let E be the minimal injective cogenerator of the category of R-modules. We give a classification of modules which are reflexive with respect to E. The result is that a module M is reflexive with respect to E if and only if M has a finitely generated submodule S such that M/S is artinian and if R/I is a complete semilocal ring where I = ann(M ).We denote by Ω the maximal spectrum of R, and we let E = m∈Ω E R (R/m) be the minimal injective cogenerator in the category of R-modules.∨∨ is an isomorphism we say that M is (Matlis) reflexive. We note that for any M , the map M → M ∨∨ is an injection. From this it is easy to conclude that ann(M ) = ann(M ∨ ). If S ⊂ R is a multiplicative set and the canonical mapIf R is a local ring we letR denote its completion. If M is finitely generated we note thatR ⊗ R M ∼ =M (the completion of M ). We writeR ⊗ R M ∼ =M = M to mean that M →R ⊗ R M ∼ =M is an isomorphism.We note that if m ∈ Ω and M is a finitely generated R-module M , then Hom R (M, E(R/m)) = 0 if and only if ann(M ) ⊂ m.

We show that over a right coherent ring all pure injective left modules have flat covers. Then using recent work of Auslander and Buchweitz we show that left modules of finite flat dimension over right coherent rings also have flat covers.

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In [4], Tȃrnȃuceanu described the finite groups G having exactly |G| − 1 cyclic subgroups. In [1], the authors used elementary methods to completely characterize those finite groups G having exactly |G| − ∆ cyclic subgroups for ∆ = 2, 3, 4 and 5. In this paper, we prove that for any ∆ > 0 if G has exactly |G| − ∆ cyclic subgroups, then |G| ≤ 8∆ and therefore the number of such G is finite. We then use the computer program GAP to find all G with exactly |G| − ∆ cyclic subgroups for ∆ = 1, . . . , 32. Date