defazajx
2020-12-21
Answered

Let R be a commutative ring. Show that R[x] has a subring isomorphic to R.

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joshyoung05M

Answered 2020-12-22
Author has **97** answers

Let R be a commutative rng and consider R[x]

Define:$\varphi :R\to R\left[x\right]$ by $r\mapsto r$

Here, clearly we can say that$\varphi$ is one-to-one and homomorphism

Now,$\varphi \left(R\right)$ is subsring of R[x] since it is the imag of a homomorphism

Then$\varphi \left(R\right)$ is subsring of R[x] isomorphic to R

Define:

Here, clearly we can say that

Now,

Then

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Find matrix of linear transformation

A linear transformation

$T:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2}$

is given by

$T\left(i\right)=i+j$

$T\left(j\right)=2i-j$

A linear transformation

is given by

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Consider the matrix transformation $T:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$ defined by

$T(x,y,z)=(\u20134x+3y+z,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x\u20135y,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}6y+7z).$

Find the standard matrix for $T$, and use that matrix to find $T(1,\u20133).$.

$T(x,y,z)=(\u20134x+3y+z,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x\u20135y,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}6y+7z).$

Find the standard matrix for $T$, and use that matrix to find $T(1,\u20133).$.

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Find

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Let $\mathbf{A}$ be the algebra given by the following multiplication table

$\begin{array}{ccccc}\cdot & 0& 1& 2& 3\\ 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 1\\ 2& 0& 0& 1& 2\\ 3& 0& 1& 2& 3\end{array}$

I need to prove that the variety generated by $\mathbf{A}$ is exactly the variety of commutative semigroups satisfying ${x}^{3}\approx {x}^{4}$.

$\begin{array}{ccccc}\cdot & 0& 1& 2& 3\\ 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 1\\ 2& 0& 0& 1& 2\\ 3& 0& 1& 2& 3\end{array}$

I need to prove that the variety generated by $\mathbf{A}$ is exactly the variety of commutative semigroups satisfying ${x}^{3}\approx {x}^{4}$.