Abstract
Let $R = \oplus_{n \in \Z} R_{n}$ be a strongly graded ring of type $\Z$ and $R_{0}$ is a prime Goldie ring. It is shown that the following three conditions are equivalent: (i) $R_{0}$ is a $\Z$-invariant Krull ring, (ii) $R$ is a Krull ring and (iii) $R$ is a graded Krull ring. We completely describe all $v$-invertible $R$-ideals in $Q$, where $Q$ is a quotient ring of $R$.
Highlights
Let R = ⊕n∈ZRn be a strongly graded ring of type Z, that is, RnRm = Rn+m for all n, m ∈ Z, where Z is the ring of integers and R0 is a prime Goldie ring with quotient ring Q0 and R0 ⊂ Q0
Let C0 = {c0 ∈ R0 | c0 is regular} is a regular Ore set of R and Qg = RC0−1, the quotient ring of R at C0, which is of the form Qg = ⊕n∈ZQ0Rn (Q0Rn = RnQ0)
It follows that Qg = Q0[X, X−1, σ], a skew Laurent polynomial ring over Q0, where σ is an automorphism of Q0 and X is a unit in Qg with X ∈ R1
Summary
Let I = I0R be a graded right R-ideal in Qg. (1) (R : I0R)l = R(R0 : I0)l and is a graded left R-ideal in Qg. A non-empty set F of graded right ideals of R is called a graded right Gabriel topology on R if the following two conditions are satisfied: a. Let G = G0R be a graded right ideal of R and F ∈ F such that rn−1 ·G ∈ Fg = {F : graded essential right ideal of R | (R : rn−1 · F )l = R for any rn ∈ Rn and n ∈ Z}.
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