Maximal Ideal in Simple Ring: Understanding the Relationship Between N and R/N

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Homework Statement


how that N is a maximal ideal in a ring R if and only if R/N is a simple ring. that is it is nontrivial and has no proper nontrivial ideals.


Homework Equations





The Attempt at a Solution


I don't know how to start. Please help.
 
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Hint: if you have an ideal in R/N, what do you get by taking its inverse image under the quotient map?
 
it is R?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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