# mmc%20rep%20of%20mcg

by Yu ·

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- Elementary Questions about a Matrix Let \[A=\begin{bmatrix} -5 & 0 & 1 & 2 \\ 3 &8 & -3 & 7 \\ 0 & 11 & 13 & 28 \end{bmatrix}.\] (a) What is the size of the matrix $A$? (b) What is the third column of $A$? (c) Let $a_{ij}$ be the $(i,j)$-entry of $A$. Calculate $a_{23}-a_{31}$. […]
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- The Preimage of Prime ideals are Prime Ideals Let $f: R\to R'$ be a ring homomorphism. Let $P$ be a prime ideal of the ring $R'$. Prove that the preimage $f^{-1}(P)$ is a prime ideal of $R$. Proof. The preimage of an ideal by a ring homomorphism is an ideal. (See the post "The inverse image of an ideal by […]
- Non-Example of a Subspace in 3-dimensional Vector Space $\R^3$ Let $S$ be the following subset of the 3-dimensional vector space $\R^3$. \[S=\left\{ \mathbf{x}\in \R^3 \quad \middle| \quad \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, x_1, x_2, x_3 \in \Z \right\}, \] where $\Z$ is the set of all integers. […]
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- Order of the Product of Two Elements in an Abelian Group Let $G$ be an abelian group with the identity element $1$. Let $a, b$ be elements of $G$ with order $m$ and $n$, respectively. If $m$ and $n$ are relatively prime, then show that the order of the element $ab$ is $mn$. Proof. Let $r$ be the order of the element […]