We are a group of Finnish second semester undergrad students who are trying to solve old exercises from this linear algebra class for the benefit of our whole semester and even all prospective students (aka TeXing the solutions for future generations and putting them on our Dropbox). So far so good, this one exercise really leaves us clueless (probably due to non of us quite understanding how to handle modules) and we hope somebody who knows "non linear" Algebra can teach us by solving this with explainations so we don't lose more time just on this one exercise ( we spent heartfelt 2-3 hours already trying to figure something, but we haven't got anything substantial which isn't just a mere guess) :
This is the "evil" exercise:
7
a) Let $R$ be a PID, $M$ a free $R$ module and let $S$ be a $R$-submodule with $\{0\} \ne S \subset M$. Let $(x_{1},\dots,x_{n})$ be a $R$-basis of $M$ and let $t_{1},\ldots,t_{m}\in R$, with $t_{1}|\cdots|t_{m} \ne 0$ so that $(t_{1}x_{1},\cdots,t_{m}x_{m})$ is a $R$-basis of $S$. If we now let $$F=\bigl\{f_{1}(y_{1})+\cdots+f_{k}(y_{k});k\ge 1, f_{1},...,f_{k}\in \mathrm{Hom}(M,R), \ y_{1},\ldots,y_{k}\in S\bigr\}.$$ Then we can show that $F=t_{1}R$
b) Let more concretely be $R=\mathbf{Z}$, $M= \mathbf{Z}^{3}$, $S= \mathbf{Z}a_{1}+\mathbf{Z}a_{2}+\mathbf{Z}a_{3}$. Putting $a_{1}=(1,2,3)$, $a_{2}=(4,5,6)$, $a_{3}=(7,8,9)$. Compute $t_{1}>0, \ldots , t_{m} > 0$ in (a)
c) Show that $M/S$ is isomorphic to $(\mathbf{Z}/3\mathbf{Z}) \times \mathbf{Z}$
When we asked our prof where he takes the exercises from, he told us he takes them from a book called "Algebra" by Artin. We couldn't find it in the book itself, so excuse us for not giving a direct reference. Our professor puts $30$% of pure Algebra stuff in the exercises and it is obvious that they are his favourite exercises (so if you want a good grade you need to know this material). We sincerely hope that a seasoned algebraist takes pity/empathy on us and excuse ourselves for the English mistakes and our mathematical incapability.