Reading: Eisenbud: Sections 1.4, 2.4.
Exercises: Eisenbud: 2.17, 1.2, 1.4.
Exercise A: (a) Let M be a Noetherian R-module and f: M
→ M be an R-module map. Show that if f is surjective, then it
is an isomorphism.
(b) Now replace the assumptions as follows: M is Artinian and f is
injective. Show that f is an isomorphism.
[Hint: Iterate f and try to get a sequence of submodules
or quotient modules from that.]
Exercise B: Let k be a field and R a finitely generated
k-algebra. Show that the following conditions are equivalent:
(i) R is an Artinian ring;
(ii) R is a finite k-algebra, i.e., R is finite-dimensional
as a vector space over k.
[Hint: To prove (i) implies (ii), use the structure
theorem on Artinian rings to reduce the problem to the local ring
case. A more general form of Hilbert's Nullstellensatz, which we
will study later, states that the residue field is a finite
extension of k. Deduce that R has a finite length as a k-module.]
Last modified: (2019-10-01 21:42:51 CDT)