6310-2023-08

Let \(k\) be the field with 2 elements. Identify all the maximal ideals in the ring \(R=k[x,y]/(xy,x^2+y+1)\).

Suppose \(R\) is a commutative ring with 1 and \(x, y \in R\) satisfy \(xy = 0\) and \((x, y) = R\) (i.e. the ideal generated by \(x\) and \(y\) is the whole ring). Prove that the \(R\)-module homomorphism

sending \(f\in R\) to its image in \(R/(x)\), and in \(R/(y)\), is an isomorphism.

Suppose \(R=\mathbb {Q}[x,y]\) is a polynomial ring. Compute the \(\mathbb {Q}\)-vector space dimension of \[ \operatorname {Tor}^R_1\left (\operatorname {Ext}^1_R(R/(x^2),R/(y^2)),R/(y)\right ). \]

If some power of a linear map \(A:\mathbb {C}^n\to \mathbb {C}^n\) is the identity (i.e. \(A\) has finite order), show that \(A\) is diagonalizable, i.e. there is a change of basis matrix \(B\) such that \(BAB^{-1}\) is diagonal.

Find four non-isomorphic \(\mathbb {Z}\)-modules (i.e. abelian groups) \(M\) that each fit into an exact sequence of the form: