TRIANGULAR MATRIX REPRESENTATIONS OF SKEW MONOID RINGS

Let R be a ring and S a u.p.-monoid. Assume that there is a monoid homomorphism &#945; : S &#8594; Aut (R). Suppose that &#945; is weakly&#12288;rigid and lR(Ra) is pure as a left ideal of R for every element a &#8712; R. Then the skew monoid ring R*S induced by &#945; has the same triangulating dimension as R. Furthermore, if R is a PWP ring, then so is R*S.</p

Special precovers and preenvelopes of complexes

The notion of an $\mathcal{L}$ complex (for a given class of $R$-modules $\mathcal{L}$) was introduced by Gillespie: a complex $C$ is called $\mathcal{L}$ complex if $C$ is exact and $\Z_{i}(C)$ is in $\mathcal{L}$ for all $i\in \mathbb{Z}$. Let $\widetilde{\mathcal{L}}$ stand for the class of all $\mathcal{L}$ complexes. In this paper, we give sufficient condition on a class of $R$-modules such that every complex has a special $\widetilde{\mathcal{L}}$-precover (resp., $\widetilde{\mathcal{L}}$-preenvelope). As applications, we obtain that every complex has a special projective precover and a special injective preenvelope, over a coherent ring every complex has a special FP-injective preenvelope, and over a noetherian ring every complex has a special $\widetilde{\mathcal{GI}}$-preenvelope, where $\mathcal{GI}$ denotes the class of Gorenstein injective modules.Comment: 12 page

FP-GR-INJECTIVE MODULES

In this paper, we give some characterizations of FP-grinjective R-modules and graded right R-modules of FP-gr-injective dimension at most n. We study the existence of FP-gr-injective envelopes and FP-gr-injective covers. We also prove that (1) (⊥gr-FI, gr-FI) is a hereditary cotorsion theory if and only if R is a left gr-coherent ring, (2) If R is right gr-coherent with FP-gr-id(RR) ≤ n, then (gr-FIn, gr-F n⊥) is a perfect cotorsion theory, (3) (⊥gr-FIn, gr-FIn) is a cotorsion theory, where gr-FI denotes the class of all FP-gr-injective left R-modules, gr-FIn is the class of all graded right R-modules of FP-gr-injective dimension at most n. Some applications are given

Stability of Gorenstein flat categories with respect to a semidualizing module

In this paper, we first introduce $\mathcal {W}_F$-Gorenstein modules to establish the following Foxby equivalence: \xymatrix@C=80pt{\mathcal {G}(\mathcal {F})\cap \mathcal {A}_C(R) \ar@[r]^{C\otimes_R-} & \mathcal {G}(\mathcal {W}_F) \ar@[l]^{\textrm{Hom}_R(C,-)}} where $\mathcal {G}(\mathcal {F})$, $\mathcal {A}_C(R)$ and $\mathcal {G}(\mathcal {W}_F)$ denote the class of Gorenstein flat modules, the Auslander class and the class of $\mathcal {W}_F$-Gorenstein modules respectively. Then, we investigate two-degree $\mathcal {W}_F$-Gorenstein modules. An $R$-module $M$ is said to be two-degree $\mathcal {W}_F$-Gorenstein if there exists an exact sequence \mathbb{G}_\bullet=\indent ...\longrightarrow G_1\longrightarrow G_0\longrightarrow G^0\longrightarrow G^1\longrightarrow... in $\mathcal {G}(\mathcal {W}_F)$ such that $M \cong$ \im(G_0\rightarrow G^0) and that $\mathbb{G}_\bullet$ is Hom$_R(\mathcal {G}(\mathcal {W}_F),-)$ and $\mathcal {G}(\mathcal {W}_F)^+\otimes_R-$ exact. We show that two notions of the two-degree $\mathcal {W}_F$-Gorenstein and the $\mathcal {W}_F$-Gorenstein modules coincide when R is a commutative GF-closed ring.Comment: 18 page

A Note on Osofsky-Smith Theorem

A famous result of B.Osofsky says that a ring R is semisimple artinian if and only if every cyclic left R-module is injective. The crucial point of her proof was to show that such a ring has finite uniform dimension. In [7], B.Osofsky and P.F.Smith proved more generally that a cyclic module M has finite uniform dimension if every cyclic subfactor of M is an extending module. Extending modules have been studied extensively in recent years and many generalizations have been considered by many authors (see, for examples, [1-4, 6, 8, 9]). Lopez-Permouth, Oshiro and Tariq Rizvi in [6] introduced the concepts of extending modules and (quasi-)continuous mod-ules relative a given left R-module X. Let S be the class of all semisimple left R-modules and all singular left R-modules. We say a left R-module N is S-extending if N is X-extending for any X 2 S. Every extending left R-module is S-extending but the converse is not true. Exploiting the techniques of [7] we prove the following result: Let M be a cyclic left R-module. Assume that all cyclic subfactors of M are S-extending. Then M satisfies ACC on direct summands. As a corollary we show that if cyclic left R-module M is extending and all cyclic subfactors of M are S-extending, then M has finite uniform dimension. Throughout this note we write A ·e B (AjB) to denote that A is an essential submodule (a direct summand) of B. A left R-module M is called singular if, for every m 2 M, the anni-hilator l(m) of m is an essential left ideal of R. Lemma 1 ([4, 4.6]). The following are equivalent for a left R-module M. (1) M is singular. (2) M » = L/K for a left R-module L and K ·e L. Let M,X be left R-modules. Define the family A(X,M) = fA µM j9Y µ X, 9f 2 Hom(Y,M), f(Y) ·e Ag. Consider the propertie