Ring $4\mathbb{R}$

Question

I have to calculate $\mathbb{R} / 4\mathbb{R}$ which is the quotient ring and I was wondering what is $4\mathbb{R}$? I think that $4\mathbb{R}=\mathbb{R}$.

Answer 1

$4\mathbb R=\{4u \mid u\in \mathbb R\}\subset \mathbb R$

But reciprocally $\forall v\in\mathbb R$ then $4\left(\dfrac v4\right)\in 4\mathbb R$ since $\frac v4$ is a real number, so $\mathbb R\subset 4\mathbb R$

Thus both are equal $\mathbb R=4\mathbb R$.

And $\mathbb R/4\mathbb R$ is reduced to only $1$ class.

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In $\mathbb R/4\mathbb R$ two elements $x,y$ are equal if we can find an element $a\in 4\mathbb R$ such that $x-y=a$.

So since $r=\dfrac{x-y}4$ is a real number for any pair of reals $(x,y)$ then $a=4r$ satisfies the requirement above, making all elements in $\mathbb R/4\mathbb R$ equal to each other.

The cardinal of this space is $1$ and we can choose $0$ as a representative for instance.

We find back the equality between $\mathbb R$ and $4\mathbb R$ by saying that every $x$ is in relation with $0$ (i.e. $x-0=a$ or simply $x=a$)