Let \(R\) be a principal ideal domain and \(M\) a finitely generated \(R\)-module. Then \(M\cong R^r\oplus R/(d_1)\oplus\cdots\oplus R/(d_s)\) for nonzero nonunits \(d_1\mid d_2\mid\cdots\mid d_s\) (invariant factors), and \(M\cong R^r\oplus\bigoplus_iR/(p_i^{e_i})\) for primes \(p_i\) (elementary divisors); the rank \(r\), the ideals \((d_j)\), and the multiset of ideals \((p_i^{e_i})\) are determined by \(M\). The Smith normal form of a presentation matrix of \(M\) computes the invariant factors. For \(R=\ZZ\) the theorem classifies finitely generated abelian groups. For \(R=F[x]\) with \(F\) a field, a finite-dimensional \(F\)-vector space with an operator \(T\) is an \(F[x]\)-module on which \(x\) acts as \(T\), and the theorem gives the canonical forms of \(T\).
Structure theorem for finitely generated modules over a PID (Tag PR-UVUS6)
Let \(R\) be a principal ideal domain and \(M\) a finitely generated \(R\)-module. Then there are an integer \(r\geq0\), nonzero nonunits \(r_1 \divides r_2 \divides \cdots\divides r_m\) of \(R\), and primes \(p_1,\ldots,p_s\) of \(R\), not necessarily distinct, with exponents \(e_i\geq1\), such that \[ M \cong R^r\oplus \bigoplus_{i=1}^m R/(r_i) \qquad\text{and}\qquad M \cong R^r\oplus \bigoplus_{i=1}^s R/(p_i^{e_i}). \] The first is the invariant factor decomposition and the second the elementary divisor decomposition. The rank \(r\) is unique, the \(r_i\) are unique up to units, and the \(p_i^{e_i}\) are unique up to units and reordering.
Invariant factor decomposition (Tag D-SS34F)
Let \(G\) be a finitely generated abelian group, and for \(n\geq 1\) let \(C_n\) denote the cyclic group of order \(n\). An invariant factor decomposition of \(G\) is an isomorphism \[ G \cong \ZZ^r \times \prod_{k=1}^m C_{n_k} \] with \(r,m\geq 0\) and integers \(n_1,\ldots,n_m\geq 2\) such that \(n_1 \divides n_2\divides \cdots \divides n_m\). The integers \(n_1,\ldots,n_m\) are the invariant factors of \(G\).
Every finitely generated abelian group \(G\) has an invariant factor decomposition, the direct product of the free abelian group \(\ZZ^r\) and finitely many finite cyclic groups. The integers \(r\) and \(m\) and the invariant factors \(n_1,\ldots,n_m\) are uniquely determined by \(G\).
Elementary divisor decomposition (Tag D-JQNJQ)
Let \(G\) be a finitely generated abelian group, and for \(n\geq 1\) let \(C_n\) denote the cyclic group of order \(n\). An elementary divisor decomposition of \(G\) is an isomorphism \[ G \cong \ZZ^r \times \prod_{k=1}^m C_{p_k^{e_k}} \] with \(r,m\geq 0\), primes \(p_1,\ldots,p_m\) that need not be distinct, and exponents \(e_1,\ldots,e_m\geq 1\). The prime powers \(p_1^{e_1},\ldots,p_m^{e_m}\) are the elementary divisors of \(G\).
Every finitely generated abelian group \(G\) has an elementary divisor decomposition. The integer \(r\) and the list \(p_1^{e_1},\ldots,p_m^{e_m}\), up to reordering, are uniquely determined by \(G\).
Invariant factors of a module over a PID (Tag FD-JTWOB)
Let \(R\) be a principal ideal domain and \(M\) a finitely generated \(R\)-module. By the structure theorem, there are an integer \(r\geq 0\) and nonzero nonunits \(r_1,\ldots,r_m\in R\) with \[ M\cong R^r\oplus\bigoplus_{i=1}^m R/(r_i) \qquad\text{and}\qquad r_1 \divides r_2 \divides \cdots \divides r_m, \] and \(r_1,\ldots,r_m\) are unique up to units. The elements \(r_1,\ldots,r_m\) are the invariant factors of \(M\).
Elementary divisors of a finite abelian group (Tag FD-H764S)
Let \(G\) be a finite abelian group. By the elementary divisor decomposition, there are primes \(p_1,\ldots,p_m\), not necessarily distinct, and integers \(\alpha_1,\ldots,\alpha_m\geq 1\) with \[ G \cong \bigoplus_{i=1}^m \ZZ/p_i^{\alpha_i} \ZZ, \] and the list \(p_1^{\alpha_1},\ldots,p_m^{\alpha_m}\) is unique up to order. These prime powers are the elementary divisors of \(G\).
Converting between elementary divisors and invariant factors (Tag PR-TLPVU)
Let \(G\) be a finite abelian group with \(G\cong\bigoplus_{i=1}^s \ZZ/m_i\ZZ\) for integers \(m_i\geq 2\), not necessarily distinct.
The elementary divisors of \(G\) are the prime powers \(p^e\), \(e\geq1\), occurring in the factorizations \(m_i=\prod_p p^{e_{p,i}}\), listed with multiplicity over all \(i\); that is, \(G\cong\bigoplus_{i}\bigoplus_{p} \ZZ/p^{e_{p,i}}\ZZ\).
For each prime \(p\), list the exponents of the \(p\)-power elementary divisors in decreasing order, \(e_{p,1}\geq e_{p,2}\geq\cdots\), padding with zeros. Put \(d_j\coloneqq\prod_p p^{e_{p,j}}\) and let \(t\) be the largest \(j\) with \(d_j\neq 1\). Then \(d_t\divides d_{t-1}\divides\cdots\divides d_1\) and \[ G\cong \ZZ/d_t\ZZ\oplus\cdots\oplus\ZZ/d_1\ZZ \] is the invariant factor decomposition of \(G\).
By the Chinese remainder theorem, \(\ZZ/m\ZZ\cong\bigoplus_p\ZZ/p^{e_p}\ZZ\) for \(m=\prod_p p^{e_p}\), which gives (a). For (b), the same theorem gives \(\ZZ/d_j\ZZ\cong\bigoplus_p\ZZ/p^{e_{p,j}}\ZZ\), so \(\bigoplus_j\ZZ/d_j\ZZ\) has the same elementary divisors as \(G\) and is isomorphic to \(G\). Since \(e_{p,j+1}\leq e_{p,j}\) for every \(p\), \(d_{j+1}\divides d_j\).
Let \(G=\ZZ/12\ZZ\oplus\ZZ/18\ZZ\oplus\ZZ/5\ZZ\). Since \(12=2^2\cdot3\) and \(18=2\cdot3^2\), the elementary divisors are \(2^2, 2, 3^2, 3, 5\). The largest power of each prime gives \(d_1=2^2\cdot3^2\cdot5=180\), and the remaining powers give \(d_2=2\cdot3=6\). Hence \(G\cong\ZZ/6\ZZ\oplus\ZZ/180\ZZ\), with \(6\divides180\).
The characteristic polynomial is the product of the invariant factors (Tag L-VDLNM)
Let \(k\) be a field, let \(A\in\Mat_n(k)\), and let \(f_1\divides f_2\divides\cdots\divides f_n\) be the invariant factors of \(A\): the monic diagonal entries of the Smith normal form of \(xI-A\) over \(k[x]\), some of which may equal \(1\). Then \[ \det(xI-A)=\prod_{j=1}^nf_j(x). \]
The Smith normal form is \(P(xI-A)Q\) with \(P,Q\) invertible over \(k[x]\), so \(\det P\) and \(\det Q\) are nonzero constants and \(\det(xI-A)\) is a nonzero constant multiple of \(\prod_jf_j\). Both polynomials are monic, so they are equal.
The minimal polynomial is the invariant factor of highest degree (Tag L-Y5KNM)
Let \(k\) be a field, let \(A\in\Mat_n(k)\), and let \(f_1\divides f_2\divides\cdots\divides f_n\) be the invariant factors of \(A\): the monic diagonal entries of the Smith normal form of \(xI-A\) over \(k[x]\). Then the minimal polynomial of \(A\) is \[ \min_A(x)=f_n(x). \]
With \(x\) acting as \(A\), the \(k[x]\)-module \(k^n\) is isomorphic to \(\bigoplus_{j=1}^nk[x]/(f_j)\). A polynomial \(g\) satisfies \(g(A)=0\) if and only if \(g\) annihilates this module, that is, \(f_j\divides g\) for every \(j\); since \(f_j\divides f_n\) for all \(j\), this holds if and only if \(f_n\divides g\). So \(f_n\) is the monic generator of the ideal of polynomials vanishing at \(A\).
Elementary divisors are the minimal polynomials of the Jordan blocks (Tag L-W5S2W)
Let \(k\) be a field and let \(A\in\Mat_n(k)\) have characteristic polynomial that splits over \(k\), so that \(A\) is similar to its Jordan canonical form \(\JCF(A)\). Then the elementary divisors of \(A\), the prime-power factors in the decomposition of the \(k[x]\)-module \(k^n\) with \(x\) acting as \(A\), are the minimal polynomials \((x-\lambda)^s\) of the Jordan blocks \(J_s(\lambda)\) of \(\JCF(A)\), listed with multiplicity.
A Jordan block \(J_s(\lambda)\) is the matrix of multiplication by \(x\) on \(k[x]/((x-\lambda)^s)\) in the basis \((x-\lambda)^{s-1},\ldots,(x-\lambda),1\), and its minimal polynomial is \((x-\lambda)^s\). So the block decomposition of \(\JCF(A)\) is a decomposition \(k^n\cong\bigoplus k[x]/((x-\lambda)^s)\), one summand for each block, and the elementary divisors are unique up to order.