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  • Ring Homomorphisms and Radical IdealsRing Homomorphisms and Radical Ideals Let $R$ and $R'$ be commutative rings and let $f:R\to R'$ be a ring homomorphism. Let $I$ and $I'$ be ideals of $R$ and $R'$, respectively. (a) Prove that $f(\sqrt{I}\,) \subset \sqrt{f(I)}$. (b) Prove that $\sqrt{f^{-1}(I')}=f^{-1}(\sqrt{I'})$ (c) Suppose that $f$ is […]
  • 12 Examples of Subsets that Are Not Subspaces of Vector Spaces12 Examples of Subsets that Are Not Subspaces of Vector Spaces Each of the following sets are not a subspace of the specified vector space. For each set, give a reason why it is not a subspace. (1) \[S_1=\left \{\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3 \quad \middle | \quad x_1\geq 0 \,\right \}\] in […]
  • A Prime Ideal in the Ring $\Z[\sqrt{10}]$A Prime Ideal in the Ring $\Z[\sqrt{10}]$ Consider the ring \[\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}\] and its ideal \[P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}.\] Show that $p$ is a prime ideal of the ring $\Z[\sqrt{10}]$.   Definition of a prime ideal. An ideal $P$ of a ring $R$ is […]
  • The Range and Nullspace of the Linear Transformation $T (f) (x) = x f(x)$The Range and Nullspace of the Linear Transformation $T (f) (x) = x f(x)$ For an integer $n > 0$, let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis. Let $T : \mathrm{P}_n \rightarrow \mathrm{P}_{n+1}$ be the map defined by, […]
  • 5 is Prime But 7 is Not Prime in the Ring $\Z[\sqrt{2}]$5 is Prime But 7 is Not Prime in the Ring $\Z[\sqrt{2}]$ In the ring \[\Z[\sqrt{2}]=\{a+\sqrt{2}b \mid a, b \in \Z\},\] show that $5$ is a prime element but $7$ is not a prime element.   Hint. An element $p$ in a ring $R$ is prime if $p$ is non zero, non unit element and whenever $p$ divide $ab$ for $a, b \in R$, then $p$ […]
  • A Recursive Relationship for a Power of a MatrixA Recursive Relationship for a Power of a Matrix Suppose that the $2 \times 2$ matrix $A$ has eigenvalues $4$ and $-2$. For each integer $n \geq 1$, there are real numbers $b_n , c_n$ which satisfy the relation \[ A^{n} = b_n A + c_n I , \] where $I$ is the identity matrix. Find $b_n$ and $c_n$ for $2 \leq n \leq 5$, and […]
  • Find the Inverse Matrix Using the Cayley-Hamilton TheoremFind the Inverse Matrix Using the Cayley-Hamilton Theorem Find the inverse matrix of the matrix \[A=\begin{bmatrix} 1 & 1 & 2 \\ 9 &2 &0 \\ 5 & 0 & 3 \end{bmatrix}\] using the Cayley–Hamilton theorem.   Solution. To use the Cayley-Hamilton theorem, we first compute the characteristic polynomial $p(t)$ of […]
  • Exponential Functions are Linearly IndependentExponential Functions are Linearly Independent Let $c_1, c_2,\dots, c_n$ be mutually distinct real numbers. Show that exponential functions \[e^{c_1x}, e^{c_2x}, \dots, e^{c_nx}\] are linearly independent over $\R$. Hint. Consider a linear combination \[a_1 e^{c_1 x}+a_2 e^{c_2x}+\cdots + a_ne^{c_nx}=0.\] […]

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