extensionofscalars

Extension of scalars is a fundamental concept in abstract algebra, particularly in the study of modules and fields. It refers to the process of taking a module or an algebra over a ring and creating a new module or algebra over a larger ring. This is typically achieved by tensoring the original module with a ring homomorphism from the original ring to the new ring.

Let R be a ring and M be a left R-module. If we have another ring S and a ring homomorphism f: R -> S, we can construct a left S-module M_S. This new module is defined as M_S = S tensor_R M, where the tensor product is taken over the ring R. The action of an element s in S on an element m in M_S is given by s * (r tensor_m m') = (sr) tensor_m m', where r is in R and m' is in M.

The operation of extension of scalars allows us to view an R-module M as an S-module when S is an extension of R. This is particularly useful when R is a subring of S. In such cases, the homomorphism f is simply the inclusion map. A common application is when R is a field and S is an extension field of R. An R-vector space can then be naturally viewed as an S-vector space. This process is also crucial in algebraic geometry and number theory, where it is used to study geometric objects and number fields over larger rings or fields.