Let R be a Gorenstein ring.We prove that if I is an ideal of R such that R/I is a semi-simple ring,then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical.In addition,we prove that if R→S is a homomorphism of rings and SE is an injective cogenerator for the category of left S-modules,then the Gorenstein flat dimension of S as a right R-module and the Gorenstein injective dimension of E as a left R-module are identical.We also give some applications of these results.
Let Λ and Γ be left and right Noetherian rings and Λ U a generalized tilting module with Γ = End( Λ U ). For a non-negative integer k, if Λ U is (k - 2)-Gorenstein with the injective dimensions of Λ U and U Γ being k, then the socle of the last term in a minimal injective resolution of Λ U is non-zero.
让螞是 Artinian 代数学和 F Ext 螞 1 的添加剂 subbifunctor (? ， ?) 有足够的 projectives 和 injectives。我们证明在 F 扩展下面关上的现代派的螞的使二元化的亚变种 F 几乎切开了序列。让 T 是在现代派的螞和 S 的一个 F-cotilting 模块在螕 = 结束(T) 上的一个 cotilting 模块。然后 Hom (? ， T ) ，在鈯 ? S 在鈯 T 和几乎裂口序列导致在 F 几乎切开的序列之间的两重性增加螕 S = Hom 螞((F) ， T ) 。让螞是 F-Gorenstein 代数学， T 一个强壮的 F-cotilting 模块和 0 鈫 ? A 鈫 ? B 鈫 ? C 鈫 ? 在鈯 ? F T 的 0 一 F 几乎切开的顺序。如果 S 的 injective 尺寸作为一个 gT 模块等于 d，那么 C？(惟厘米 ? d 惟 d DTrA *)* ，在此(-)*= Hom (， T ) 。另外，如果 A 的 F-injective 尺寸等于 d。关键词F几乎切开了序列-几乎裂口序列- F-Gorenstein 代数学先生( 2000 )题目分类 16G70 - 16E05 部分为高等教育(资助号码 20060284002 )的博士节目由专业化研究资金支持了，中国(资助号码 10771095 )的国家自然科学基础和中国的江苏省的国家自然科学基础(资助没有。BK2007517 ）
In this paper, we first introduce the notion of generalized k-syzygy modules, and then give an equivalent characterization that the class of generalized k-syzygy modules coincides with that ofω-k-torsionfree modules. We further study the extension closure of the category consisting of generalized k-syzygy modules. Some known results are obtained as corollaries.
Chong-hui HUANG & Zhao-yong HUANG Department of Mathematics, Nanjing University, Nanjing 210093, China
