1. ## Free modules

In each case determine if M is an free A-module. I such a case give a basis.

a) $\displaystyle A=Z,M=\{(0,3m+2n,m+n):m,n \in \mathbb{Z} \}$, as submodule of $\displaystyle \mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}$

b) A is a conmutative ring, $\displaystyle M=M_n(A)$, with the action $\displaystyle (aM)ij=am_{ij},$ for all $\displaystyle a \in A, M \in M_n(A)$

2. Originally Posted by roporte
In each case determine if M is an free A-module. I such a case give a basis.

a) $\displaystyle A=Z,M=\{(0,3m+2n,m+n):m,n \in \mathbb{Z} \}$, as submodule of $\displaystyle \mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}$

b) A is a conmutative ring, $\displaystyle M=M_n(A)$, with the action $\displaystyle (aM)ij=am_{ij},$ for all $\displaystyle a \in A, M \in M_n(A)$
both are free. in part a) the basis is $\displaystyle \{(0,3,1), (0,2,1)\}$ and in part b) the basis is $\displaystyle \{e_{ij}: \ 1 \leq i,j \leq n \},$ where $\displaystyle e_{ij} \in M_n(A)$ is defined to have 1 in the (i,j) entry and 0 everywhere else.