let $r$ be a right artinian ring or a perfect commutative‎‎ring‎. ‎let $m$ be a noncosingular self-generator $sum$-lifting‎‎module‎. ‎then $m$ has a direct decomposition $m=oplus_{iin i} m_i$‎,‎where each $m_i$ is noetherian quasi-projective and each‎‎endomorphism ring $end(m_i)$ is local‎.

Let $(R,fm)$ be a Gorenstein local ring and$M,N$ be two finitely generated modules over $R$. Nagel proved that if $M$ and $N$ are inthe same even liaison class, thenone has $H^i_{fm}(M)cong H^i_{fm}(N)$ for all $iIn this paper, we provide a short proof to this result.

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...

By a well known theorem of K. Morita, any equivalence between full module categories over rings R and S, are given by a bimodule RPS , such that RP is a finitely generated projective generator in R-Mod and S = EndR(P ). There are various papers which describe equivalences between certain subcategories of R-Mod and S-Mod in a similar way with suitable properties of RPS . Here we start from the o...

In this paper certain injectivity conditions in terms of extensions of monomorphisms are considered. In particular, it is proved that a ring R is a quasi-Frobenius ring if and only if every monomorphism from any essential right ideal of RR into R (N) R can be extended to RR. Also, known results on pseudo-injective modules are extended. Dinh raised the question if a pseudo-injective CS module is...

We introduce the concepts of lifting modules and (quasi-)discrete modules relative to a given left module. We also introduce the notion of SSRS-modules. It is shown that (1) if M is an amply supplementedmodule and 0→N ′ →N →N ′′ → 0 an exact sequence, then M isN-lifting if and only if it isN ′-lifting andN ′′-lifting; (2) ifM is a Noetherianmodule, then M is lifting if and only if M is R-liftin...

We establish braided tensor equivalences among module categories over the twisted quantum double of a finite group defined by an extension of a group G by an abelian group, with 3-cocycle inflated from a 3-cocycle on G. We also prove that the canonical ribbon structure of the module category of any twisted quantum double of a finite group is preserved by braided tensor equivalences. We give two...

1.1. We will define a D-module of differential operators on a smooth stack and construct a symbol map when the stack is good, but first we need to give definition of a D-module on a stack. D-modules are local for the smooth topology, so one’s naive guess for the definition of a D-module on a smooth Artin stack is correct. That is, a (left or right) D-module M on X is the assignment for each U π...

Let g be a finite dimensional complex semi-simple Lie algebra with Weyl group W and simple reflections S. For I ⊆ S let g I be the corresponding semi-simple subalgebra of g. Denote by W I the Weyl group of g I and let w • and w I • be the longest elements of W and W I , respectively. In this paper we show that the answer to Kostant's problem, i.e. whether the universal enveloping algebra surjec...