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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 1997 • Question 6

Question

Math 6310 • Algebra Fall 1997 • Question 6

Let \(A\) and \(B\) be finite rings with the same number of elements. Assume that \(n\) is an integer \(>1\) and \(x^n=x\) for all \(x\in A\) and \(y^n=y\) for all \(y\in B\).

  1. If \(n=6\), prove that \(A\) and \(B\) are isomorphic.
  2. If \(n=4\), prove by an example that \(A\) and \(B\) need not be isomorphic.

Raw LaTeX

Let \(A\) and \(B\) be finite rings with the same number of elements. Assume that \(n\) is an integer
\(>1\) and \(x^n=x\) for all \(x\in A\) and \(y^n=y\) for all \(y\in B\).
\begin{enumerate}
\item[(a)] If \(n=6\), prove that \(A\) and \(B\) are isomorphic.
\item[(b)] If \(n=4\), prove by an example that \(A\) and \(B\) need not be isomorphic.
\end{enumerate}

Topic Tags

Ideals and Quotient Rings