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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2012 • Question 3

Question

Math 6310 • Algebra Fall 2012 • Question 3

Let \(R\) be a commutative ring with 1 and \(M_n(R)\) the ring of \(n\times n\) matrices with coefficients in \(R\). Prove that every ideal of \(M_n(R)\) is of the form \(M_n(I)\), for some ideal \(I\) of \(R\).

Raw LaTeX

Let \(R\) be a commutative ring with 1 and \(M_n(R)\) the ring of \(n\times n\) matrices with
coefficients in \(R\). Prove that every ideal of \(M_n(R)\) is of the form \(M_n(I)\), for some ideal
\(I\) of \(R\).

Topic Tags

Ideals and Quotient Rings Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization Polynomial Rings and Prime Ideals