On fully left bounded left Noetherian rings

Prime M-Ideals, M-Prime Submodules, M-Prime Radical and M-Baer's Lower Nilradical of Modules

“…It follows immediately that, the ring R satisfies condition H if and only if every finitely generated left R-module is finitely annihilated. We note the stronger result due to Krause [13] that if R is left Noetherian, then there is a one-to-one correspondence between isomorphism classes of indecomposable injective left R-modules and prime ideals of R if and only if R is a left fully bounded ring (see [9,Theorem 8.12] for a proof). In [1,Theorem 6.7], Beachy shown that Gabriel's correspondence can be extended to M -injective modules, provided that Hom R (M, X) = 0 for all modules X in σ[M ].…”

Ore Extensions over Total Valuation Rings

“…We can define fully right bounded similarly, and a ring is said to be fully bounded if it is fully left bounded and fully right bounded. In [1] Asano introduced the concept of boundedness in order to study the order theory and Krause characterized fully left bounded left Noetherian rings in terms of indecomposable injective modules [12,Theorem 3.5].…”

Topologies on the torsion-theoretic spectrum of a noncommutative ring

“…Parts (l)-(4) follow from [11,Proposition 3.1]. As for part (5), ass (N) £ ass (M) by (3). Conversely, assume that τ e ass (M).…”