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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2005 • Question 4

Question

Math 6310 • Algebra Fall 2005 • Question 4

Let \(S\) be a multiplicative subset of commutative ring \(A\) not containing 0. Let \(P\) be a maximal element in the set of ideals of \(A\) whose intersection with \(S\) is empty. Show that \(P\) is prime.

Raw LaTeX

Let \(S\) be a multiplicative subset of commutative ring \(A\) not containing 0.
Let \(P\) be a maximal element in the set of ideals of \(A\) whose intersection with \(S\) is empty.
Show that \(P\) is prime.

Topic Tags

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