$(m,d)$-内射盖与Gorenstein $(m,d)$-平坦模

$(m,d)$-Injective Covers and Gorenstein $(m, d)$-Flat Modules

中文关键词:  $(m,d)$-内射盖  Gorenstein $(m,d)$-平坦模  Gorenstein $(m,d)$-内射模  强Gorenstein $(m,d)$-平坦模

英文关键词:$(m, d)$-injective cover  Gorenstein $(m, d)$-flat module  Gorenstein $(m, d)$-injecitve module  strongly Gorenstein $(m, d)$-flat module

中文摘要:

      研究了$(m,d)$-内射$R$-模作成的类是(预)盖类的条件,证明了$(m,d)$-凝聚环上的每一个左$R$-模都具有$(m,d)$-内射盖.在此基础上,又引入研究了Gorenstein $(m,d)$-平坦模和Gorenstein $(m,d)$-内射模,证明了$(m,d)$-凝聚环上的左$R$-模$M$是Gorenstein$(m,d)$-平坦模的充分必要条件是它的特征模$M^{+}$是Gorenstein $(m,d)$-内射模.推广了Goresntein平坦模和Goresntein $n$-平坦模上的一些结果.

英文摘要:

      We consider the conditions under which the class of $(m, d)$-injective $R$-modules is (pre)covering. It is shown that every left $R$-module over a left $(m, d)$-coherent ring has an $(m, d)$-injective cover. Moreover, the classes of Gorenstein $(m, d)$-flat modules and Gorenstein $(m, d)$-injecitve modules are introduced and studied. For a right $(m, d)$-coherent ring $R$, we prove that a left $R$-module $M$ is Gorenstein $(m, d)$-flat if and only if $M^{+}$ is Gorenstein $(m, d)$-injective as a right $R$-module. Some results on Gorenstein flat modules and Gorenstein $n$-flat modules are generalized.