Injective modules and localization in noncommutative Noetherian rings

Abstract:

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