Answer to Question #141830 in Abstract Algebra for anjali

Question

If M,N are R-module then M x N is also R-module

Answer 1

M x N the following operations are set:

$(m_1,n_1)+(m_2,n_2)=(m_1+m_2,n_1+n_2)\\ r*(m_1,n_1)=(r*m_1,r*n_1)\\ r\in R,\ m_1,m_2\in M,n_1,n_2\in N$

Since operations on M x N are "distributed" into two parts in M and N separately, it is quite obvious that M x N is an R-module. For example, let's check the commutativity:

$(m_1,n_1)+(m_2,n_2)=(m_1+m_2,n_1+n_2)=\\ =(m_2+m_1,n_2+n_1)=(m_2,n_2)+(m_1,n_1)$