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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2005 • Question 5

Question

Math 6310 • Algebra Fall 2005 • Question 5

Let \((R,m)\) be a local ring (i.e. a commutative ring with a unique maximal ideal \(m\)). Let \(M\) be a finitely generated \(R\)-module with \(mM=M\). Show from first principles that \(M=\{0\}\).

Raw LaTeX

Let \((R,m)\) be a local ring (i.e. a commutative ring with a unique maximal ideal \(m\)).
Let \(M\) be a finitely generated \(R\)-module with \(mM=M\). Show from first principles that
\(M=\{0\}\).

Topic Tags

Ideals and Quotient Rings Modules over PIDs Polynomial Rings and Prime Ideals Prime and Maximal Ideals