If R is a ring and I is a right ideal of R then I is called faithful if R - I is a faithful right R-module, i.e. if { r ∊ R: Rr⊆ I} = (0). I is called irreducible [ 1 ] provided that if J1 and J2 are right ideals such that J1 ∩ J2 = I, then J1 or J2 = I. Let N(I){ r ∊ R: rI⊆ I} and [ I: a ] = { r ∊ R: ar⊆ I} for a ∊ R. We write (a)r for [(0): a ].
Throughout we assume that R is a left noetherian ring, not necessarily commutative. R-modules are left modules, [M] denotes the isomorphism class and E(M) the injective hull of a module M.The (left) spectrum of R, denoted Spec R, is taken to be the set of isomorphism classes of indecomposable injective modules. We denote by G(R) the directed graph whose set of vertices is the set Spec R and such that if [KJ ,
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