Injective modules and localization in noncommutative Noetherian rings

Abstract

Let S \mathfrak {S} be a semiprime ideal in a right Noetherian ring R and C ( S ) = { c ∈ R | [ c + S \mathcal {C}(\mathfrak {S}) = \{ c \in R|[c + \mathfrak {S} regular in R / S } R/\mathfrak {S}\} . We investigate the following two conditions: ( A ) C ( S ) ({\text {A}})\;\mathcal {C}(\mathfrak {S}) is a right Ore set in R. ( B ) C ( S ) ({\text {B}})\;\mathcal {C}(\mathfrak {S}) is a right Ore set in R and the right ideals of R S {R_{\mathfrak {S}}} , the classical right quotient ring of R w.r.t. C ( S ) \mathcal {C}(\mathfrak {S}) are closed in the J ( R S ) J({R_{\mathfrak {S}}}) -adic topology. The main results show that conditions (A) and (B) can be characterized in terms of the injective hull of the right R-module R / S R/\mathfrak {S} . The J-adic completion of a semilocal right Noetherian ring is also considered.