Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2022 • Question 4

Question

Math 6310 • Algebra Fall 2022 • Question 4

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.

Suppose that \(R\) is a commutative ring and \(0\to M\xrightarrow{\alpha}N\xrightarrow{\beta}P\to0\) 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.

Topic Tags

Abelian Groups and Classification of Finite Abelian Groups Exact Sequences of Modules Hom Functors and Module Homomorphisms Polynomial Rings and Ring Homomorphism Kernels