We introduce the notion of right pre-resolutions (quasi-resolutions) for noncommutative isolated singularities, which is a weaker version quasi-resolutions introduced by Qin-Wang-Zhang. prove that noetherian bounded below and locally finite graded algebra with injective dimension 2 are always Morita equivalent. When we restrict to quadric hypersurfaces, hypersurface, singularity, admits pre-res...

Isbell and Semadeni [Trans. Amer. Math. Soc. 107 (1963)] proved that every infinite-dimensional $1$-injective Banach space contains a hyperplane is $(2+\varepsilon )$-injective for $\varepsilon \gt 0$, yet not $2$-injective, rem

We show that if a compact, connected, and oriented $n$-manifold $M$ without boundary admits non-constant non-injective uniformly quasiregular self-map, then the dimension of real singular cohomology ring $H^*(M;\Bbb{R})$ is bounded from above by $2^n$. This positive answer to dynamical counterpart Bonk-Heinonen conjecture on bound for quasiregularly elliptic manifolds. The proof based an interm...

LetR be a ring andM a rightR-module with S= End(MR). The moduleM is called almost principally quasi-injective (or APQ-injective for short) if, for any m∈M, there exists an S-submodule Xm of M such that lMrR(m) = Sm ⊕ Xm. The module M is called almost quasiprincipally injective (or AQP-injective for short) if, for any s∈ S, there exists a left ideal Xs of S such that lS(ker(s)) = Ss ⊕ Xs. In thi...