I'm reading about tensor product of modules, there's a theorem in the book that leaves parts of the proof to the reader. I've attached the file, I didn't put this in HW section because first of all I thought this question was more advanced to be posted in there and also because I want to discuss something else with people on this forum. I'm thinking what happens when a simple tensor becomes zero? I mean suppose that we have [itex]x\otimes y = 0[/itex]. Does it mean that x or y must be zero? or it's possible that one of them nonzero but still their tensor product turns out to be zero? If possible, please check my proof in the file as well. I know I'm asking too much, but please help me as quickly as possible because tomorrow I'll have a conference about tensors, the professor is also the head of the math. department of our university and I'm the only under-graduate student in his class. so I'm very determined to have a successful conference tomorrow in the class and I need your help very much guys.