Directions. Although you may refer to books for definitions and standard theorems, searching for solutions to these written exercises is not permitted. You may not seek help from others.
Bear in mind that rings are always assumed to have a multiplicative identity, and a homomorphism of rings is always assumed to carry the multiplicative identity of its domain to that of its target. Recall that if is a ring, the term -algebra indicates, by definition, a pair where is a ring and is a ring homomorphism.
If is a field and an integer, it has been explained in this course that an -module structure on extending its usual -module structure amounts to the same thing as an matrix in . Show that the -modules corresponding to two given matrices and are isomorphic if and only if and are similar over .
For a prime it is a fact that the sequence of coefficients in the -adic expansion of a -adic integer that is rational must eventually repeat.
Find the -adic expansion of the rational number .
Find the expansion of 2/3 as a real “decimal” computed in base .
If is a ring, employ a suitable universal mapping property to state what is meant by an abstract product of a non-empty collection of -algebras. Then prove that such always exist.
Employ a suitable universal mapping property to state what is meant by a direct limit for a non-empty collection of rings indexed by a directed set with a suitable collection of connecting ring homomorphisms. Use this definition in carrying out the following task.
Let be a domain, its field of fractions, and a prime ideal in . Let denote the subring of consisting of those elements of representable as where with , and for let denote the subring of consisting of those elements of representable as for some integer . Note that when is a multiple of , there is an inclusion homomorphism from to . Show that is a direct limit of .
Let be a field, finite-dimensional vector spaces over , and the dual of . Prove that together with the bilinear map defined by is a tensor product of and over .