TopicS covered: Finitely Generated Free-modules over a P.I.D. [Finitely generated free modules: examples and its Properties, properties of submodules of finitely generated free modules, Finite abelian groups and cyclic modules. ]
FACULTY OF SCIENCE
RAM JI DEPARTMENT : MATHEMATICS
PANDEY COURSE : M.A./M.Sc. (Mathematics)
Assistant Professor SEMESTER : II (Second)
PAPER : First (MAT2TH02) CONTACT ADDRESS: Department of Mathematics, References: Ewing Christian College, 1. Ramji Lal: Algebra (Volume 2), Springer Prayagraj – 211003 Publication.
2. C. Musili: Introduction to Rings and Module, PHONE: Narosa Publication. +91 6394787490 3. D.S. Dummit & R. M. Foote: Abstract Algebra,
John Wiley Publication. EMAIL: 4. F. W. Anderson & K. R. Fuller: Rings and [email protected] Categories of Modules. [email protected]
Finitely Generated Free Modules over a P.I.D. MODULE THEORY
OBJECTIVE In this study material you will learn the following: Finitely generated free modules : Definition, examples and non-examples. Properties of Finitely generated free modules. Existence of basis of a submodule of a finitely generated free modules over a P.I.D. Properties of submodules of finitely generated free modules over a P.I.D. and finite abelian groups. Free abelian groups and cyclic modules.
Definition: (Finitely generated free module) A R-module M is said to be finitely generated free module if and only if it has a basis consisting of finite number of elements.
Examples: 1. The R-module , where R is any ring with identity, is finitely generated free module as { , ,…, } is a basis of . where = (0,0,…,1,…,0) (1 is in the place). 2. ℤ as ℤ -module is finitely generated but not free. (Justify)
[Hint: ℤ is a finite set and so it must be generated by whole ℤ , which shows ℤ is finitely generated] 3. The collection of all sequences in a ring R is a R-module, where R is a ring with identity viz free but not finitely generated. (Justify)
[Hint: Find a countable basis for above R-module as taken in Example 1 (simply extend the n-tuples to infinite sequence)]
Proposition: A finitely generated free R-module having basis consisting of n elements is isomorphic to .
Proof: Let M be a free R-module with basis ={ , ,…, } consisting of n elements. As we know ={ , ,…, } is a basis of , so define a map : ⟶ by, ( ) = ;∀ = 1,2,…, then, as M is free, so it must satisfies homomorphism extension property. So, can be uniquely extended to a homomorphism :̅ ⟶ such that ̅ ( ) = ( ) = ; ∀ = 1,2,…, Claim: ̅ is an isomorphism.
For, = ∈ ̅ then ̅ ( ) =0
⟹ ̅ =0 ̅ ̅ ⟹ ( ) = 0 (as f is a homomorphism)
M.A./M.Sc. (Mathematics) Sem II RAM JI PANDEY 1
Finitely Generated Free Modules over a P.I.D. MODULE THEORY