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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2008 • Question 3

Question

Math 6310 • Algebra Spring 2008 • Question 3

Let \(R\) be a commutative ring with 1. Assume that \(R\) satisfies the ascending chain condition. Let \(I\) be an ideal generated by an infinite sequence of elements \(x_1,x_2,\ldots \) in \(R\). Show that \(I\) is finitely generated.

Raw LaTeX

Let \(R\) be a commutative ring with 1. Assume that \(R\) satisfies the ascending chain condition.
Let \(I\) be an ideal generated by an infinite sequence of elements \(x_1,x_2,\ldots\) in \(R\). Show
that \(I\) is finitely generated.

Topic Tags

Exact Sequences of Modules Ideals and Quotient Rings Polynomial Rings and Prime Ideals