# ring-theory-eye-catch

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• Vector Space of Polynomials and a Basis of Its Subspace Let $P_2$ be the vector space of all polynomials of degree two or less. Consider the subset in $P_2$ $Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},$ where \begin{align*} &p_1(x)=1, &p_2(x)=x^2+x+1, \\ &p_3(x)=2x^2, &p_4(x)=x^2-x+1. \end{align*} (a) Use the basis $B=\{1, x, […] • A Ring is Commutative if Whenever$ab=ca$, then$b=c$Let$R$be a ring and assume that whenever$ab=ca$for some elements$a, b, c\in R$, we have$b=c$. Then prove that$R$is a commutative ring. Proof. Let$x, y$be arbitrary elements in$R$. We want to show that$xy=yx$. Consider the […] • Are Coefficient Matrices of the Systems of Linear Equations Nonsingular? (a) Suppose that a$3\times 3$system of linear equations is inconsistent. Is the coefficient matrix of the system nonsingular? (b) Suppose that a$3\times 3$homogeneous system of linear equations has a solution$x_1=0, x_2=-3, x_3=5$. Is the coefficient matrix of the system […] • Primary Ideals, Prime Ideals, and Radical Ideals Let$R$be a commutative ring with unity. A proper ideal$I$of$R$is called primary if whenever$ab \in I$for$a, b\in R$, then either$a\in I$or$b^n\in I$for some positive integer$n$. (a) Prove that a prime ideal$P$of$R$is primary. (b) If$P$is a prime ideal and […] • Prove$\mathbf{x}^{\trans}A\mathbf{x} \geq 0$and determine those$\mathbf{x}$such that$\mathbf{x}^{\trans}A\mathbf{x}=0$For each of the following matrix$A$, prove that$\mathbf{x}^{\trans}A\mathbf{x} \geq 0$for all vectors$\mathbf{x}$in$\R^2$. Also, determine those vectors$\mathbf{x}\in \R^2$such that$\mathbf{x}^{\trans}A\mathbf{x}=0$. (a)$A=\begin{bmatrix} 4 & 2\\ 2& […]
• A Relation of Nonzero Row Vectors and Column Vectors Let $A$ be an $n\times n$ matrix. Suppose that $\mathbf{y}$ is a nonzero row vector such that $\mathbf{y}A=\mathbf{y}.$ (Here a row vector means a $1\times n$ matrix.) Prove that there is a nonzero column vector $\mathbf{x}$ such that $A\mathbf{x}=\mathbf{x}.$ (Here a […]
• Non-Abelian Group of Order $pq$ and its Sylow Subgroups Let $G$ be a non-abelian group of order $pq$, where $p, q$ are prime numbers satisfying $q \equiv 1 \pmod p$. Prove that a $q$-Sylow subgroup of $G$ is normal and the number of $p$-Sylow subgroups are $q$.   Hint. Use Sylow's theorem. To review Sylow's theorem, check […]
• Determine the Number of Elements of Order 3 in a Non-Cyclic Group of Order 57 Let $G$ be a group of order $57$. Assume that $G$ is not a cyclic group. Then determine the number of elements in $G$ of order $3$.   Proof. Observe the prime factorization $57=3\cdot 19$. Let $n_{19}$ be the number of Sylow $19$-subgroups of $G$. By […]