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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2006 • Question 5

Question

Math 6310 • Algebra Fall 2006 • Question 5

Suppose \(R\) is a commutative ring with identity. Let \(J\) denote the intersection of all maximal ideals in \(R\). Suppose \(M\) is a finitely-generated \(R\)-module such that \[ JM:=\{jm\mid j\in J\text { and }m\in M\}=\{0\}. \] Prove that \(M=\{0\}\).

Raw LaTeX

Suppose \(R\) is a commutative ring with identity. Let \(J\) denote the intersection of all maximal
ideals in \(R\). Suppose \(M\) is a finitely-generated \(R\)-module such that
\[
JM:=\{jm\mid j\in J\text{ and }m\in M\}=\{0\}.
\]
Prove that \(M=\{0\}\).

Topic Tags

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