Tensor product with a lattice

Question

Let $\omega_1,\omega_2\in\Bbb C$ be $\Bbb R$-linearly independent.

Let $\Gamma=\Bbb{Z}\omega_1\oplus \Bbb Z\omega_2$. What is $\Gamma\otimes \Bbb R$? I imagine we are taking a tensor product as $\Bbb Z$-algebras? In which case is this not just $\Bbb C$?

Answer 1

This might help

The tensor product of two dimensioned spaces A, B is the simplest bilinear mapping M which for basis elements a,b implies M(a,b) is a basis element of the tensor product.

Apply the bilinearity to your problem :))