PID Modules, The Structure of a Finitely Generated Module

The Structure of a Finitely Generated Module

Remember, a cyclic submodule is generated by a single element, e.g. R*c for some c in M, and is isomorphic to R/H, where H is the annihilator of c. In this case H is generated by some principal element x, so we may write the cyclic module R*c as R/x.

When R = Z, we are talking about Z modules, or abelian groups. In this case a cyclic module is the integers, or the integers mod n for some positive integer n.

Let Q be the quotient module M/M_T. Note that torsion free maps to torsion free under this homomorphism, thus Q is torsion free, and finitely generated. The previous theorem tells us Q is free.

Some of the generators of M lie in M_T, and generate M_T. The remaining generators span Q. This places a bound on the rank of Q. In particular, the rank is finite, since M is finitely generated.

The quotient module Q, and its rank, are isomorphic invariants of M. In other words, the rank of Q is not arbitrary; it is determined by the structure of M.

Select a basis for Q, then let b₁ b₂ b₃ etc be the preimage of these basis elements in M. It doesn't matter how you choose the preimages, as long as they induce a basis for Q. Each b_i is outside of M_T, and is torsion free. They span a finitely generated torsion free submodule of M, which is free. With no kernel, this submodule is isomorphic to Q. It is a copy of Q lying in M; let's just call it Q. Therefore M = Q*M_T.

We understand Q, a free R module whose rank is bounded by the number of generators outside of M_T. To complete the description of M, we only need characterize the torsion submodule M_T.

Let R be a pid and let M be a finitely generated torsion R module.

For each prime element p in R, let W_p be the set of elements in M whose order is some power of p. (Remember that the order of c in M is p^k if the ideal p^k*R is the annihilator of c.) Let's verify that W_p is a submodule.

If p^k kills c, then it kills yc. The order of yc is a factor of p^k, which is still a power of p. That takes care of scaling, and addition is also straightforward. If p^k kills c and p^j kills d, then p^j+k kills c+d, and the order of c+d is a factor of p^j+k, which is a power of p.

The order of c in M depends only on the structure of M, as an R module, hence the various submodules W_p are well defined. They are uniquely determined by M.

The generators of M have specific orders, each carrying a finite number of primes, and since there are finitely many generators, there are finitely many submodules W_p to consider. We want to show that M is the direct product of these submodules.

If the generators are spanned then M is spanned. Consider a generator g, having order x, where x is the product of p_i^k_i, over the relevant primes p_i. Let x_i = x/p_i^k_i. Since these values have no gcd, some linear combination s_ix_i equals 1. (This is bezout's identity.) Write 1*g as the sum of s_ix_ig. If the i^th term in the sum is multiplied by p_i^k_i, the result is s_ixg, which is 0 (since xg is 0). The i^th term is killed by a power of p, and belongs to W_p. Therefore g is spanned by our submodules. This holds for each generator, hence the finite collection of submodules W_p spans M.

Next, demonstrate linear independence. Suppose b is in W_p and in V, where V is the span of W_q for each q ≠ p. Since b is in W_p its order is a power of p. Yet b also belongs to V. Let z be the product of q_i^k_i for all q ≠ p. Note that z kills each W_q, hence z kills all of V. In particular, z kills b. The ideal that kills b contains z and some power of p. These to elements are relatively prime, hence they span 1. The order of b is 1, and b = 0. Our submodules W_p are linearly independent, and M is the direct product of these submodules.

We now need to understand the module M, having order p^k. The invariant factor theorem does this for us, hence M splits into a direct product of cyclic modules, killed by various powers of p, and the decomposition is unique. Each cyclic submodule is isomorphic to R mod p^k, for some exponent k.

When R is the integers, we're talking about the integers mod p, p², p³, p⁴, etc. Select any combination of these rings, take their direct product, and find W_p.

To summarize, a finitely generated module over a pid is the direct product of a free module whose rank is bounded by the number of generators, times a finite number of cyclic modules, which are quotients of various prime powers in R.