# Three-pieces2

• Prove the Ring Isomorphism $R[x,y]/(x) \cong R[y]$ Let $R$ be a commutative ring. Consider the polynomial ring $R[x,y]$ in two variables $x, y$. Let $(x)$ be the principal ideal of $R[x,y]$ generated by $x$. Prove that $R[x, y]/(x)$ is isomorphic to $R[y]$ as a ring.   Proof. Define the map $\psi: R[x,y] \to […] • Commutator Subgroup and Abelian Quotient Group Let$G$be a group and let$D(G)=[G,G]$be the commutator subgroup of$G$. Let$N$be a subgroup of$G$. Prove that the subgroup$N$is normal in$G$and$G/N$is an abelian group if and only if$N \supset D(G)$. Definitions. Recall that for any$a, b \in G$, the […] • If the Quotient by the Center is Cyclic, then the Group is Abelian Let$Z(G)$be the center of a group$G$. Show that if$G/Z(G)$is a cyclic group, then$G$is abelian. Steps. Write$G/Z(G)=\langle \bar{g} \rangle$for some$g \in G$. Any element$x\in G$can be written as$x=g^a z$for some$z \in Z(G)$and$a \in \Z$. Using […] • Orthogonal Nonzero Vectors Are Linearly Independent Let$S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$be a set of nonzero vectors in$\R^n$. Suppose that$S$is an orthogonal set. (a) Show that$S$is linearly independent. (b) If$k=n$, then prove that$S$is a basis for$\R^n$. Proof. (a) […] • There is Exactly One Ring Homomorphism From the Ring of Integers to Any Ring Let$\Z$be the ring of integers and let$R$be a ring with unity. Determine all the ring homomorphisms from$\Z$to$R$. Definition. Recall that if$A, B$are rings with unity then a ring homomorphism$f: A \to B$is a map […] • Find Values of$a, b, c$such that the Given Matrix is Diagonalizable For which values of constants$a, b$and$c$is the matrix $A=\begin{bmatrix} 7 & a & b \\ 0 &2 &c \\ 0 & 0 & 3 \end{bmatrix}$ diagonalizable? (The Ohio State University, Linear Algebra Final Exam Problem) Solution. Note that the […] • Centralizer, Normalizer, and Center of the Dihedral Group$D_{8}$Let$D_8$be the dihedral group of order$8$. Using the generators and relations, we have $D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle.$ (a) Let$A$be the subgroup of$D_8$generated by$r$, that is,$A=\{1,r,r^2,r^3\}\$. Prove that the centralizer […]