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While tensor products are quite prolific in commutative algebra, even some of their most basic properties remain relatively unknown. We explore one of these properties, namely a tensor's torsion. In particular, given any finitely generated modules, M and N over a ring R, the tensor product $M\otimes_R N$ almost always has nonzero torsion unless one of the modules M or N is free. Specifically, we look at which rings guarantee nonzero torsion in tensor products of non-free modules over the ring. We conclude that a specific subclass of one-dimensional Gorenstein rings will have this property.
Adviser: Roger Wiegand and Tom Marley