I am really struggling to understand several parts of the definition of tensor product given in my lecture notes:
Definition of the tensor product *Denote by L the free A-module with a basis consisting of elements l$_{m,n}$ indexed by elements of M x N. So an arbitrary element of L is a finite sum $\Sigma$ a$_i$l$_{m_i,n_i}$ with a$_i$$\in$ A, m$_i$$\in$ M and n$_i$$\in$ N. Let K be the A-submodule of L generated by elements: l$_{m_1+m_2,n}$ - l$_{m_1,n}$ - l$_{m_2,n}$; l$_{m,n_1+n_2}$ - l$_{m,n_1}$ - l$_{m,n_2}$ ; l$_{am,n}$ - al$_{m,n}$; l$_{m,an}$ - al$_{m,n}$ (for all a $\in$ A, m $\in$ M and n $\in$ N ). Denote T = L/K. The image of l$_{m,n}$ in T , i.e. the coset l$_{m,n}$ + K is usually denoted by m $\otimes$ n. Since L is generated by l$_{m,n}$, the module T is generated by m $\otimes$ n, i.e. T = {$\Sigma$ a$_i$m$_i$$\otimes$n$_i$ : a$_i$ $\in$ A;m$_i$ $\in$ M; n$_i$ $\in$ N }
These satisfy relations:
(m$_1$ + m$_2$) $\otimes$ n = m$_1$ $\otimes$ n + m$_2$ $\otimes$ n; (am) $\otimes$ n = a(m $\otimes$ n); m $\otimes$ (n$_1$ + n$_2$) = m $\otimes$ n$_1$ + m $\otimes$ n$_2$;
m $\otimes$ (an) = a(m $\otimes$ n)
(for all a $\in$ A, m $\in$ M and n $\in$ N ).
The module T is denoted M $\bigotimes$$_A$ N and is called the tensor product of
M and N over A.*
In the notes it is stated, as if obvious, that K is an A-submodule of L and that T = L/K is a module - but I just don't see how.
I also don't understand why the relations are satisfied - I try plugging in some arbitrary values l$_{m_1,n_1}$ etc. to try and prove the relations, but I have no idea how to manipulate these since to me they are just indexed l's which I can't do anything with.
Really appreciate any guidance you can give me
