Pure-injective modules
Pure-injective modules
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DOI：
10.1017/s0017089500001853
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发表时间：
1973-09
影响因子：
0.5
通讯作者：
Syed M. Fakhruddin
中科院分区：
文献类型：
--
作者：
Syed M. Fakhruddin
本文证明了具有单位元的交换环上的纯内射模是具有诱导模结构的n个表示模的乘积的和项，其中n个表示模是参照圈群T来理解的。利用类似的技巧，还证明了当R-模是循环模的一个乘积的子模，并且是相同乘积的交换群的被加数时，它的基础群是纯内射的。所有考虑的环是可交换的单位和所有的模是酉的。设Mod-R是环R上的模范畴。Mod-R中的正合序列0 → A → B → C → 0是纯正合的，如果对任意的N，0 → A <$N → B <$N → C <$N → 0是正合的。模M是纯内射的，如果它相对于Mod-R中的纯正合序列类具有内射性质。一个模块P是FP（FP表示），如果它是一个FP生成的自由模块与一个FP生成的内核的镜像。一个模M是紧的，如果它带有一个Hausdorff紧拓扑使得M是一个拓扑R-模。设T表示圆群--以整数为模的真实的数群--设X* 表示模X的对偶模Hom（X，T）。
Abstract It is proved that a pure-injective module over a commutative ring with unity is a summand of a product of duals of finitely presented modules, where duals are to be understood with reference to the circle group T, with induced module structures. Using similar techniques, it is also shown that an R-module has its underlying group pure-injective precisely when it is a submodule of a product of duals of cyclic modules and also a summand as abelian group of the same product. All rings considered are commutative with unity and all modules are unitary. Let Mod-R be the category of modules over a ring R. An exact sequence 0 → A → B → C → 0 in Mod-R is pure-exact if, for any N in Mod-R, 0 → A⊗N → B⊗N → C⊗N → 0 is exact. A module M is pure-injective if it has the injective property relative to the class of pure-exact sequences in Mod-R. A module P is FP (finitely presented) if it is the image of a finitely generated free module with a finitely generated kernel. A module M is compact if it carries a Hausdorff compact topology so that M is a topological R-module. Let T denote the circle group—the group of real numbers modulo the integers—and let X* denote the dual module Homℤ(X, T) of the module X.