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 2 N, and 1 < r1 j r2 j · · · j 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 2 A, denote Oa := fr 2 R j ra = 0g. 1. Oa is an ideal of R. 2. At := fa 2 A j Oa 6= 0g is a submodule, called the torsion submodule of A. Ora ⊃ Oa and Oa+b ⊃ Oa \Ob for r 2 R and a; b 2 A. 3. For a 2 A there is an isomorphism of left modules R=Oa ' Ra = fra j r 2 Rg: 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 2 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 2 R, let A(p) := fa 2 A j a has order a power of pg: 1. A(p) ≤ A for each prime p 2 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 2 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 j r2 j · · · j 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 2 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]..