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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2009 • Question 7

Question

Math 6310 • Algebra Fall 2009 • Question 7

Let \(R\) be a commutative ring, and let \(S\) be a multiplicative subset of \(R\) (\(1\in S\), and if \(s\) and \(t\) are in \(S\), then \(st\) is in \(S\)). Show that an ideal \(I\) that is maximal with the property that \(S\cap I=\emptyset \) is prime.

Raw LaTeX

Let \(R\) be a commutative ring, and let \(S\) be a multiplicative subset of \(R\) (\(1\in S\), and
if \(s\) and \(t\) are in \(S\), then \(st\) is in \(S\)). Show that an ideal \(I\) that is maximal with the
property that \(S\cap I=\emptyset\) is prime.

Topic Tags

Ideals and Quotient Rings Polynomial Rings and Prime Ideals Prime and Maximal Ideals