Exercise 22, § 4

A module M over a principal ideal domain is said to have finite rank if there exists an integer $m \geq 0$ with the following property: every finitely generated submodule of M is contained in a submodule generated by m elements or fewer. The smallest integer $m$ with this property is called the rank of M. a) Show that if M has finite rank, then every quotient module has finite rank, at most equal to that of M. b) For a torsion-free module M, the notion of rank defined above coincides with that defined in II, p. 315. c) Show that if M has finite rank then every submodule of M has finite rank, at most equal to that of M. d) Let M be a $\pi$-primary module such that $M(\pi) = M$ (in the notation of VII, p. 7). Show that if M has finite rank then it is finitely generated. e) Let M be a $\pi$-primary module with no element of infinite height. Show that if M has finite rank then it is finitely generated, and the notion of rank coincides with that defined in

Ex. 8. (Notice that if $M(\pi)$ is finitely generated then so is $M$, by proving that the heights of elements of $M(\pi)$ in $M$ are bounded ; use Ex. 5 of VII, p. 55 to show this.) f) Let $M$ be a $\tau$-primary module of finite rank $r$, and let $M_s$ denote the submodule consisting of all the elements of infinite height in $M$. Show that $M_s$ is a divisible module, and the direct sum of a finite number $p \leq r$ of indecomposable submodules $U_{\pi}$ (VII, p. 54, Ex. 3), and that $M$ is the direct sum of $M_0$ and a finitely generated submodule $N$ of rank $r - p$. (Notice that $M/M_0$ is finitely generated, using e) ; then apply Ex. 3 of VII, p. 54.) g) Deduce from the above that an $A$-module $M$ has finite rank $r$ if and only if the quotient module $M/T$ of $M$ by its torsion submodule $T$ has finite rank $p \leq r$, and that each $\pi$-primary component of $T$ has rank $\leq r - p$, with at least one of them having rank $r - p$. h) Let $(p_n)$ be the sequence of distinct prime numbers ; let $E$ be the direct sum of the cyclic $\mathbf{Z}$-modules $\mathbf{Z}/p_n^2\mathbf{Z}$, and let $e_n$ denote the element whose coordinates all vanish except the n-th, which is equal to the residue class of 3 mod $p_n^2$. Let $M$ be the submodule of the $\mathbf{Z}$-module $E \times \mathbf{Q}$ generated by the elements $(e_n, 1/p_n)$. Show that $M$ is a $\mathbf{Z}$-module of rank 2, that the torsion submodule $T$ of $M$ is generated by the elements $(p_n e_n, 0)$, and that $T$ is not a direct factor of $M$. (If $\bar{x}_n$ is the class of $(e_n, 1/p_n)$ mod $T$, show that for all $m \neq n$ there exists a class $\bar{y}_{nm}$ mod $T$ such that $\bar{x}_n = p_m \bar{y}_{nm}$, but that there is no element $x_n$ in the class $\bar{x}_n$ such that for all $m \neq n$ there exists $y_n \in M$ with $x_n = p_m y_{nm}$.) i) If $M$ is a finitely generated $A$-module, show that the rank of $M$ is equal to the smallest cardinal $\gamma(M)$ of any generating set for $M$. j) Let $M$ be a torsion $A$-module of finite rank. Show that if $N$ is a proper submodule of $M$ then $N$ cannot be isomorphic to $M$ (use f)).