Proving Local Rings
Not what you're looking for?
i) Show that a non-zero ring R is local (i.e. it has a unique maximal ideal) if and only if x or 1 - x is a unit, where x is a member of R.
ii) Show why or why not are the following four rings local: {a/b belongs to Q s.t. b is not divisible by a prime p}, Z[x]/(X^3), F[[x]] (where F[[x]] is the ring formed by the set of all formal power series p(x) = a_0 + (a_1)(x) + (a_2)(x^2) + ... where a_i belongs to the field F), {p(x)/q(x) belongs to Q(x) s.t. q(0) = 0}.
iii) For the previous rings that you decided are not local, slightly change their definition (without proof) such that you get a local ring.
Purchase this Solution
Solution Summary
This solution provides the detailed proofs and examples of local rings in attached .docx and .pdf formats.
Solution Preview
See the attachments.
(i) Show that a non-zero ring R is local if and only if x or 1-x is a unit, for all x in R.
Let's first prove the following proposition.
Proposition 1. A commutative ring R with 1≠0 is local if and only if the non-units of R form an ideal.
Proof: (⟹) Suppose that R is a local ring with a unique maximal ideal M. Since M is a proper ideal of R, it contains no units of R. Now let x be an element not in M. If x is not a unit, then x generates an ideal (x) which is contained in a maximal ideal N≠M. But this is impossible since M is a unique maximal ideal. Therefore, x is a unit of R. Therefore, M contains all the non-units of R.
(⟸) Suppose that R is a ring in which all the non-units form an ideal M. Then every proper ideal of R is contained in M since proper ideals cannot contain units. Thus M is a unique maximal ideal of R and therefore R is a local ring.
From Proposition 1, we immediately get this corollary.
Corollary 1. Let R be a local ring with 1≠0. Then its unique maximal ideal consists precisely of the non-units of R.
We now return to the proof.
Proposition 2. A ...
Purchase this Solution
Free BrainMass Quizzes
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Multiplying Complex Numbers
This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.