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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2002 • Question 11

Question

Math 6310 • Algebra Fall 2002 • Question 11

Let \(R\) be a an algebra over a field \(k\) of dimension \(n\). Let \(I\) be the intersection of all maximal left ideals of \(R\). Show that \(I\) is a 2-sided ideal and that \(I^n=0\). Give an example for which \(I^{n-1}\ne 0\).

Raw LaTeX

Let \(R\) be a an algebra over a field \(k\) of dimension \(n\). Let \(I\) be the intersection of all
maximal left ideals of \(R\). Show that \(I\) is a 2-sided ideal and that \(I^n=0\). Give an example
for which \(I^{n-1}\ne 0\).

Topic Tags

Ideals and Quotient Rings Prime and Maximal Ideals