TUT ALG4 1

Exercise 392.1(Artin (2011) Ex. 14.2.3).

Let $A$ be the matrix of a homomorphism $\phi :{\mathbb{Z}}^{\oplus n}\to {\mathbb{Z}}^{\oplus m}$ of free $\mathbb{Z}$-modules.

1. Prove that $\phi$ is injective iff the rank of $A$, as a real matrix, is $n$.
2. Prove that $\phi$ is surjective iff the greatest common divisor of the determinants of the $m\times m$ minors of $A$ is $1$.

---

References

Artin, M. (2011). Algebra (2. ed). Pearson Education, Prentice Hall.