4.6 Modules Over a Principal Ideal Domain

88 CHAPTER 4. MODULES

4.6 Modules over a Principal Ideal Domain

Let R be a principal ideal domain (PID), A an R-module, and F a free R-module. Ex. An ﬁnitely generated abelian group is isomorphic to

k t M M r s r ⊕ i and ⊕ rj Z Zpi Z Z i=1 j=1 for some primes pi, r, k, si, rj ∈ N, and 1 < r1 | r2 | · · · | rt. Thm 4.22. Let F be a free R-module and G a submodule of F . Then G is a free R-module and rank G ≤ rank F . Cor 4.23. If an R-module A is generated by n elements, then every submodule of A may be generated by m elements with m ≤ n. Cor 4.24. An R-module A is free iﬀ A is projective.

Lem 4.25. For a ∈ A, denote Oa := {r ∈ R | ra = 0}.

1. Oa is an ideal of R.

2. At := {a ∈ A | Oa 6= 0} is a submodule, called the torsion submodule of A.

Ora ⊃ Oa and Oa+b ⊃ Oa ∩ Ob

for r ∈ R and a, b ∈ A.

3. For a ∈ A there is an isomorphism of left modules

R/Oa ' Ra = {ra | r ∈ R}.

Remark.

1. A is a torsion module if A = At; A is torsion-free if At = 0. 2. Every free module is torsion-free. However, a non ﬁnitely generated torsion-free module may not be free (e.g. the Z-module Q).

3. For a ∈ A, if Oa = (r), then Ra ' R/Oa = R/(r) is said to be cyclic of order r. Thm 4.26. If A is ﬁnitely generated, then A is torsion-free iﬀ A is free.

Thm 4.27. If A is ﬁnitely generated, then A = At ⊕ F , where F ' A/At is a free R-module of ﬁnite rank. 4.6. MODULES OVER A PRINCIPAL IDEAL DOMAIN 89

Let us invesetigate the torsion part of A.

Lem 4.28. Let A be a torsion module. For each prime p ∈ R, let

A(p) := {a ∈ A | a has order a power of p}.

1. A(p) ≤ A for each prime p ∈ R; X 2. A = A(p). If A is ﬁnitely generated, only ﬁnitely many A(p) are nonzero. p prime

n1 nk Lem 4.29 (Chinese Remainder Theorem). If r = p1 ··· pk where pi are distinct primes in R, then k M ni R/(r) ' R/(pi ) as left R-modules. i=1 Lem 4.30. Let p ∈ R be a prime. Let A be finitely generated such that every nonzero element of A has the order a power of p. Then A is a direct sum of cyclic R-modules of orders n1 n p , ··· , p k for some n1 ≥ n2 ≥ · · · ≥ nk ≥ 1. The classification theorem of finitely generated modules over a PID is:

Thm 4.31. Let R be a PID, and A a ﬁnitely generated R-module.

1. t r M A ' R ⊕ R/(rj) j=1

where r1, ··· , rt are (not necessary distinct) nonzero nonunit elements of R such that r1 | r2 | · · · | rt. The rank r and the list of ideals (r1), ··· , (rt) are uniquely determined by A. The elements r1, ··· , rt are called the invariant factors of A. 2. k r M si A ' R ⊕ R/(pi ), i=1

where r ∈ N, p1, ··· , pk are (not necessary distinct) primes in R and s1, ··· , sk are (not s1 sk necessary distinct) positive integers. The rank r and the list of ideals (p1 ), ··· , (pk ) s1 sk are uniquely determined by A. The elements p1 , ··· , pk are called the elementary divisors of A.

Cor 4.32. Two ﬁnitely generated R-modules A and B are isomorphic iﬀ A/At and B/Bt have the same rank and A and B have the same invariant factors [resp. elementary divisors].