6310-2022-08

Find every prime ideal \(Q\) of \(R=\mathbb {Z}[x]/(x^3-3x,4+x)\) and count the number of elements in each quotient ring

\(R/Q\).

Let \(k=\mathbb {Z}/2\mathbb {Z}\) be the field with two elements and let \(R=k[x]\). Let \(M\) be the cokernel of the mapping from \(R^3\) to \(R^3\) given by the matrix \[ \begin {bmatrix} x^2&x^2&0\\ x^2&x^2+x&x+1\\ 0&0&x+1 \end {bmatrix} \]

How many elements are in \(\operatorname {Hom}_R(k[x]/(x),M)\)?

Let \(R=\mathbb {R}[x,y]\), let \(I=(x,y)\) be the ideal generated by \(x\) and \(y\) and let \(M=R/I\). Compute \(\dim _{\mathbb {R}}\operatorname {Tor}^R_i(I,M)\) for all \(i\).

Suppose that \(R\) is a commutative ring and \(0\to M\xrightarrow {\alpha }N\xrightarrow {\beta }P\to 0\) is a short exact sequence of \(R\)-modules and \(L\) is an \(R\)-module. Prove, via an explicit argument involving module homomorphisms (and without citing properties of the \(\operatorname {Hom}\) functor), that there is an exact sequence of Abelian groups: \[ 0\to \operatorname {Hom}_R(P,L)\xrightarrow {\bar {\beta }}\operatorname {Hom}_R(N,L)\xrightarrow {\bar {\alpha }}\operatorname {Hom}_R(M,L) \]

You must concretely explain how to induce the maps \(\bar {\alpha }\) and \(\bar {\beta }\) from \(\alpha \) and \(\beta \) respectively.

Let \(k\) be the field with 2 elements and let \(R=k[x]\). Identify, up to isomorphism, all \(R\)-modules \(M\) with 8 elements (that is \(|M|=8\)) and such that \(M\otimes _R (R/(x^2(x+1)))\) also has 8 elements.