Jtdylktuy

0 $begingroup$ It is a problem 10.4.10 of Dummit and Foote. Suppose $R$ is commutative and $Ncong R^n$ is a free $R-$ module of rank $n$ with $R-$ module basis $e_1,e_2,dots, e_n$ . (a) For any nonzero $R$ -module $M$ show that every element of $Motimes N$ can be written uniquely in the form $sum_{i=1}^{n}m_iotimes e_i$ where $m_iin M$ . Deduce that if $sum_{i=1}^{n}m_iotimes e_i=0$ in $Motimes N$ then $m_i=0$ for $i=1,dots,n$ . Actually I already proved the first part. Since any element in $Motimes N$ is a finite sum of simple tensors, for any ${m^j}_{j=1}^{p}subset M$ and ${n^j}_{j=1}^psubset N$ observe that begin{align*} sum_{j=1}^{p}m^jotimes n^j=sum_{j=1}^{p} sum_{i=1}^{n}m^jotimes r_i^je_i=sum_{i=1}^{n}sum_{j=1}^{p}(m^jr^j_i)otimes e_i=sum_{i=1}^{n}left( sum_{j=1}^{p}m^jr^j_irigh