# ring-theory-eye-catch

by Yu · Published · Updated

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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 […]