are (R,R)-bimodules where and By 3), we can conclude is a (R,R)-bimodule. Computing the tensor product For vector spaces, the tensor product is quickly computed since bases of of immediately determine a basis of as was mentioned above. For modules over a general (commutative) ring, not every module is free. For example, is not a free abelian group (-module). The tensor product with is given by More generally, given a presentation of some -module , that is, a number of generators together with relations the tensor product can be computed as the following cokernel: Here and the map is determined by sending some in the th copy of to (in ). Colloquially, this may be rephrased by saying that a presentation of gives rise to a presentation of This is referred to by saying that the tensor product is a right exact functor. It is not in general left exact, that is, given an injective map of -modules the tensor product is not usually injective. For example, tensoring the (injective) map given by multiplication with , with yields the zero map , which is not injective. Higher Tor functors measure the defect of the tensor product being not left exact. All higher Tor functors are assembled in the derived tensor product. Tensor product of algebras Let be a commutative ring. The tensor product of -modules applies, in particular, if and are -algebras. In this case, the tensor product is an -algebra itself by putting For example, A particular example is when and are fields containing a common subfield . The tensor product of fields is closely related to Galois theory: if, say, , where is some irreducible polynomial with coefficients in , the tensor product can be calculated as where now is interpreted as the same polynomial, but with its coefficients regarded as elements of . In the larger field , the polynomial may become reducible, which brings in Galois theory. For example, if is a Galois extension of , then is isomorphic (as an -algebra) to the Eigenconfigurations of tensors Square matrices with entries in a field represent linear maps of vector spaces, say and thus linear maps of projective spaces over If is nonsingular then is well-defined everywhere, and the eigenvectors of correspond to the fixed points of The eigenconfiguration'' of consists of points in provided is generic and is algebraically closed. The fixed points of nonlinear maps