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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2010 • Question 3

Question

Math 6310 • Algebra Spring 2010 • Question 3

Let \(R\) be a ring with 1. A left \(R\)-module \(M\) is called simple if \(M\ne 0\) and if the only submodules of \(M\) are \(M\) and 0. Show that every simple module is isomorphic to \(R/I\) for some maximal left ideal \(I\) and that \(I\) is unique if \(R\) is commutative.

Raw LaTeX

Let \(R\) be a ring with 1. A left \(R\)-module \(M\) is called \emph{simple} if \(M\ne 0\) and if the
only submodules of \(M\) are \(M\) and 0. Show that every simple module is isomorphic to \(R/I\) for
some maximal left ideal \(I\) and that \(I\) is unique if \(R\) is commutative.

Topic Tags

Ideals and Quotient Rings Modules over PIDs Polynomial Rings and Prime Ideals Prime and Maximal Ideals