We apply minimal weakly generating sets to study the existence of Add ( U R ) -covers for a uniserial module . If is right over ring , then S : = End has at most two maximal (right, left, two-sided) ideals: one set I all endomorphisms that are not injective, and other K surjective. prove if either finitely generated, or artinian, ? class covering only it closed under direct limit. Moreover, we ...

For an Artinian (n− 1)-Auslander algebra Λ with global dimension n(≥ 2), we show that if Λ admits a trivial maximal (n − 1)-orthogonal subcategory of modΛ, then Λ is a Nakayama algebra and the projective or injective dimension of any indecomposable module in modΛ is at most n− 1. As a result, for an Artinian Auslander algebra with global dimension 2, if Λ admits a trivial maximal 1-orthogonal s...

The purpose of this note is to characterize the finite Hilbert functions which force all of their artinian algebras to enjoy the Weak Lefschetz Property (WLP). Curiously, they turn out to be exactly those (characterized by Wiebe in [W i]) whose Gotzmann ideals have the WLP. This implies that, if a Gotzmann ideal has the WLP, then all algebras with the same Hilbert function (and hence lower Bett...

A module is called uniseriat if it has a unique composition series of finite length. A ring (always with 1) is called serial if its right and left free modules are direct sums of uniserial modules. Nakayama, who called these rings generalized uniserial rings, proved [21, Theorem 171 that every finitely generated module over a serial ring is a direct sum of uniserial modules. In section one we g...

 Let $R=oplus_{nin Bbb N_0}R_n$ be a Noetherian homogeneous ring with local base ring $(R_0,frak{m}_0)$, $M$ and $N$ two finitely generated graded $R$-modules. Let $t$ be the least integer such that $H^t_{R_+}(M,N)$ is not minimax. We prove that $H^j_{frak{m}_0R}(H^t_{R_+}(M,N))$ is Artinian for $j=0,1$. Also, we show that if ${rm cd}(R_{+},M,N)=2$ and $tin Bbb N_0$, then $H^t_{frak{m}_0R}(H^2_...

We find a sufficient condition that H is not level based on a reduction number. In particular, we prove that a graded Artinian algebra of codimension 3 with Hilbert function H = (h0, h1, . . . , hd−1 > hd = hd+1) cannot be level if hd ≤ 2d + 3, and that there exists a level Osequence of codimension 3 of type H for hd ≥ 2d+k for k ≥ 4. Furthermore, we show that H is not level if β1,d+2(I ) = β2,...