# Johns-Hopkins-University-exam-eye-catch

• Every Integral Domain Artinian Ring is a Field Let $R$ be a ring with $1$. Suppose that $R$ is an integral domain and an Artinian ring. Prove that $R$ is a field.   Definition (Artinian ring). A ring $R$ is called Artinian if it satisfies the defending chain condition on ideals. That is, whenever we have […]
• The Polynomial $x^p-2$ is Irreducible Over the Cyclotomic Field of $p$-th Root of Unity Prove that the polynomial $x^p-2$ for a prime number $p$ is irreducible over the field $\Q(\zeta_p)$, where $\zeta_p$ is a primitive $p$th root of unity.   Hint. Consider the field extension $\Q(\sqrt[p]{2}, \zeta)$, where $\zeta$ is a primitive $p$-th root of […]
• For Which Choices of $x$ is the Given Matrix Invertible? Determine the values of $x$ so that the matrix $A=\begin{bmatrix} 1 & 1 & x \\ 1 &x &x \\ x & x & x \end{bmatrix}$ is invertible. For those values of $x$, find the inverse matrix $A^{-1}$.   Solution. We use the fact that a matrix is invertible […]
• Independent Events of Playing Cards A card is chosen randomly from a deck of the standard 52 playing cards. Let $E$ be the event that the selected card is a king and let $F$ be the event that it is a heart. Prove or disprove that the events $E$ and $F$ are independent. Definition of Independence Events […]
• Nilpotent Element a in a Ring and Unit Element $1-ab$ Let $R$ be a commutative ring with $1 \neq 0$. An element $a\in R$ is called nilpotent if $a^n=0$ for some positive integer $n$. Then prove that if $a$ is a nilpotent element of $R$, then $1-ab$ is a unit for all $b \in R$.   We give two proofs. Proof 1. Since $a$ […]
• The Image of an Ideal Under a Surjective Ring Homomorphism is an Ideal Let $R$ and $S$ be rings. Suppose that $f: R \to S$ is a surjective ring homomorphism. Prove that every image of an ideal of $R$ under $f$ is an ideal of $S$. Namely, prove that if $I$ is an ideal of $R$, then $J=f(I)$ is an ideal of $S$.   Proof. As in the […]
• Eigenvalues of a Stochastic Matrix is Always Less than or Equal to 1 Let $A=(a_{ij})$ be an $n \times n$ matrix. We say that $A=(a_{ij})$ is a right stochastic matrix if each entry $a_{ij}$ is nonnegative and the sum of the entries of each row is $1$. That is, we have $a_{ij}\geq 0 \quad \text{ and } \quad a_{i1}+a_{i2}+\cdots+a_{in}=1$ for $1 […] • A Line is a Subspace if and only if its$y$-Intercept is Zero Let$\R^2$be the$x$-$y$-plane. Then$\R^2$is a vector space. A line$\ell \subset \mathbb{R}^2$with slope$m$and$y$-intercept$b$is defined by $\ell = \{ (x, y) \in \mathbb{R}^2 \mid y = mx + b \} .$ Prove that$\ell$is a subspace of$\mathbb{R}^2$if and only if$b = […]