Tensor/Modules

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Tensor products can also be defined for modules over the same ring. If the ring is non-commutative, we'll need to be careful about distinguishing right modules and left modules. We will write RM for a left module, and MR for a right module. If a module has both a left module structure over a ring R and a right module structure over a ring S, and in addition for every m, r, s, r(ms) = (rm)s, then we will say M is a left-right module, and will denote it by RMS. Note that every left-module is a left-right module with Z as the right ring, and vice versa. When defining the tensor, we also need to be careful about the ring: most modules can be considered as modules over several different rings. The most general form of the tensor definition is as follows: let SMR and RNK be modules, then the tensor over R is an R-bilinear operator T: M x N -> P such that for every R-bilinear operator B: M x N -> O there is a unique linear operator L: P -> O such that L o T = B. There is unique left-right module structure on P such that T is right-linear over S and left-linear over K, and if B is right-linear over S then so is L, and if B is left-linear over K then so is L. (P, T) are unique up to a unique isomorphism, and are called the "tensor space" and "tensor product" respectively. Example: Let Q and Z/p be considered as modules over the integers, Z. Let B: Q x Z/p -> M be a bilinear operator. Then b(q,i) = b(q/p, p*i) = b(q/p, 0) = 0, so every bilinear operator is identically zero. Therefore, if we define P to be the trivial module, and T to be the zero bilinear function, then we see that the properties for the tensor product are satisfied. Therefore, the tensor of Q and Z/p is {0}.

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