Is This Ring Connected?

Groups, Rings, Domains, Modules, etc, Galois theory
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Tsakanikas Nickos
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Is This Ring Connected?

#1

Post by Tsakanikas Nickos »

Let \( \displaystyle R \) be an associative ring with unity \( \displaystyle 1_{R} \) and let \( \displaystyle \mathbb{M}_{n}(R) \) be the ring of \( \displaystyle n \times n \) matrices over \( \displaystyle R \). Examine whether \( \displaystyle \mathbb{M}_{n}(R) \) is a connected ring.
Papapetros Vaggelis
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Re: Is This Ring Connected?

#2

Post by Papapetros Vaggelis »

Hello Nickos.

In general, an associative ring \(\displaystyle{\left(S,+,\cdot\right)}\) with unity \(\displaystyle{1_{S}}\) is connencted if the only

central elements \(\displaystyle{s\in S}\), having the property \(\displaystyle{s^2=s}\), are \(\displaystyle{0_{S}\,,1_{S}}\) .

Center of \(\displaystyle{\left(S,+,\cdot\right)}\) : \(\displaystyle{\,\,\,Z\,(S)=\left\{s\in S: s\cdot x=x\cdot s\,\forall\,x\in S\right\}}\) .

Obviously, \(\displaystyle{\mathbb{O}\,,I_{2}\in Z\,(\mathbb{M_{n}}(R)}\) and \(\displaystyle{\mathbb{O}^2=\mathbb{O}\,,I_{n}^2=I_{n}}\) .

It's known that :

\(\displaystyle{Z\,(\mathbb{M_{n}}(R))=\left\{r\cdot I_{n}\in\mathbb{M_{n}}(R): r\in Z\,(R)\right\}}\) .

Let \(\displaystyle{A\in\mathbb{M_{n}}(R)}\) such that \(\displaystyle{A\in Z\,(\mathbb{M_{n}}(R))}\) and \(\displaystyle{A^2=A}\) .

Since \(\displaystyle{A\in Z\,(\mathbb{M_{n}}(R))}\) , we have that \(\displaystyle{A=r\cdot I_{n}}\) for some \(\displaystyle{r\in R}\) with

\(\displaystyle{r\in Z\,(R)}\) . Then,

\(\displaystyle{A^2=A\implies \left(r\cdot I_{n}\right)\cdot \left(r\cdot I_{n}\right)=r\cdot I_{n}\implies r^2\cdot I_{n}=r\cdot I_{n}\implies r^2=r}\) .

So, if the ring \(\displaystyle{\left(R,+,\cdot\right)}\) is connected, then \(\displaystyle{\left(\mathbb{M_{n}}(R),+,\cdot\right)}\) is connected too.

On the other hand, suppose that \(\displaystyle{\left(\mathbb{M_{n}}(R),+,\cdot\right)}\) is connected.

Let \(\displaystyle{r\in R}\) such that \(\displaystyle{r\in Z\,(R)}\) and \(\displaystyle{r^2=r}\) . Then,

\(\displaystyle{r\cdot I_{n}\in Z\,(\mathbb{M_{n}}(R))}\) and

\(\displaystyle{\left(r\,I_{n}\right)^2=r^2\cdot I_{n}=r\cdot I_{n}}\). Since, \(\displaystyle{\left(\mathbb{M_{n}}(R),+,\cdot\right)}\) is connected,

we get \(\displaystyle{r\,I_{n}=\mathbb{O}\,\lor r\cdot I_{n}=I_{n}\iff r=0_{R}\,\lor r=1_{R}}\).

Therefore, the ring \(\displaystyle{\left(R,+,\cdot\right)}\) is connected.
Tsakanikas Nickos
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Posts: 314
Joined: Tue Nov 10, 2015 8:25 pm

Re: Is This Ring Connected?

#3

Post by Tsakanikas Nickos »

Thank you for your answer, Vaggelis!

Firstly, i would like to mention that "elements \( s∈S \), having the property \( s^2=s \)" are called idempotents.

Secondly, it is clear from your reply that
" An associative ring \( \displaystyle R \) with unity \( \displaystyle 1_{R} \) is connected if and only if the ring \( \displaystyle \mathbb{M}_{n}(R) \) is connected. "
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