University of Utah, Department of Mathematics Algebra 1 Qualifying Exam August 2020

Let \(R\) be a commutative Noetherian ring, and let \(\varphi :R\longrightarrow R\) be a surjective ring homomorphism. Is \(\varphi \) necessarily an isomorphism?

Prove or disprove: each element of finite order in the general linear group \(\operatorname {GL}_n(\mathbb {C})\) is diagonalizable.

Suppose \(M\) is a \(3\times 3\) matrix, with entries from the field of real numbers, such that \[ M^8=I\qquad \text {and}\qquad M^4\ne I. \] Determine the possibilities for the minimal polynomial of \(M\).

Let \(I\) be the ideal \((x,y)\) in the polynomial ring \(R:=\mathbb {C}[x,y]\), and let \(\Lambda ^2(I)\) denote the second exterior power of \(I\) in the category of \(R\)-modules. Prove or disprove: \(\Lambda ^2(I)\) is zero.

Determine all maximal ideals of the ring \(\mathbb {C}[x,y]/(x^3-x^2y,xy^2+xy+x+1)\).