Modules over a Pid
نویسنده
چکیده
Every vector space over a field K that has a finite spanning set has a finite basis: it is isomorphic to Kn for some n > 0. When we replace the scalar field K with a commutative ring A, it is no longer true that every A-module with a finite generating set has a basis: not all modules have bases. But when A is a PID, we get something nearly as good as that: (1) Every submodule of An has a basis of size at most n. (2) Every finitely generated torsion-free A-module M has a finite basis: M ∼= An for a unique n > 0. (3) Every finitely generated A-module M is isomorphic to Ad ⊕ T , where d > 0 and T is a finitely generated torsion module. We will prove this based on how a submodule of a finite free module over a PID sits inside the free module. Then we’ll learn how to count with ideals in place of positive integers.
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