Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 1995 • Question 8

Question

Math 6310 • Algebra Fall 1995 • Question 8

Let \(R\) be a commutative ring with an identity and assume that \(I\) and \(J\) are ideals in \(R\) such that \(I+J=R\). Prove that \(IJ=I\cap J\) and that \(R/IJ\) is isomorphic to the direct product \((R/I)\times (R/J)\).

Raw LaTeX

Let \(R\) be a commutative ring with an identity and assume that \(I\) and \(J\) are ideals in \(R\) such
that \(I+J=R\). Prove that \(IJ=I\cap J\) and that \(R/IJ\) is isomorphic to the direct product
\((R/I)\times(R/J)\).

Topic Tags

Chinese Remainder Theorem and Product Decompositions Ideals and Quotient Rings Polynomial Rings and Prime Ideals Tensor Products of Modules