Directions. It is intended that you work these as exercises. Although you may refer to books for definitions and standard theorems, searching for solutions to these written exercises either in books or in online references should not be required and is undesirable. If you make use of a reference other than class notes, you must properly cite that use. You may not seek help from others.
In this problem set the notation for a positive integer denotes the ring given by
Let be a commutative ring, an -module, and a group that acts on by -linear automorphisms. Show that there is one and only one way to endow with the structure of module over the group ring (of with coefficients in ) that is compatible with the given -module structure on and such that the -module scalar multiplication of any element by (i.e., by where is the canonical group homomorphism ) has the same result as does applying to via the given group action of on .
Let be a ring, and (left) -modules, the product of and , the map , and the map . Show that the pair has the property that for any -module and any pair of -linear maps with and there is one and only one -linear map such that and .
A (left) module on a commutative ring always gives rise to a bi--module with the property that for all and one has . Let be the polynomial ring where is the field of two elements, and let be the two-dimensional Cartesian space . Give an example of an -bi-module structure on for which the right multiplication by a scalar is not equal to the left multiplication.
Regarding the rings prove the following:
If , then
For each integer
For a prime let denote the field of fractions of the ring . Prove the following statements:
The set of non-units in , i.e., , is an additive subgroup of .
For any in at least one of the two elements must be in .