Specification Of Ideals Of A Product Ring

Groups, Rings, Domains, Modules, etc, Galois theory
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Tsakanikas Nickos
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Posts: 314
Joined: Tue Nov 10, 2015 8:25 pm

Specification Of Ideals Of A Product Ring

#1

Post by Tsakanikas Nickos »

Let \( \displaystyle \left\{ D_{j} \right\} _{j=1}^{k} \) be division rings and let \( \displaystyle \left\{ \mathbb{M}_{n_{j}}(D_{j}) \right\} _{j=1}^{k} \) be the respective rings of \( \displaystyle n_{j} \times n_{j} \) matrices over \( \displaystyle D_{j} \). If \( \displaystyle R = \prod_{j=1}^{k} \mathbb{M}_{n_{j}}(D_{j}) \), show that every right (left or two-sided, respectively) ideal of \( \displaystyle R \) is of the form \( \displaystyle eR \; (Re \; , \; \tilde{e}R) \), where \( \displaystyle e^{2} = e \in R \) and \( \displaystyle \tilde{e}^{2} = \tilde{e} \in Z(R) \).
Papapetros Vaggelis
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Joined: Mon Nov 09, 2015 1:52 pm

Re: Specification Of Ideals Of A Product Ring

#2

Post by Papapetros Vaggelis »

Let \(\displaystyle{j\in\left\{1,...,k\right\}}\) and \(\displaystyle{I}\) be a right(left) ideal of \(\displaystyle{\left(D_{j},+,\cdot\right)}\) .

Suppose that \(\displaystyle{I\neq \left\{0\right\}}\) . Then, there is \(\displaystyle{x\in D_{j}}\) such that \(\displaystyle{x\in I}\) and \(\displaystyle{x\neq 0}\) .

Since the ring \(\displaystyle{\left(D_{j},+,\cdot\right)}\) is a division ring, there is \(\displaystyle{x^{-1}\in D_{j}-\left\{0\right\}}\) which has the property

\(\displaystyle{x^{-1}\cdot x=1=x\cdot x^{-1}}\). Due to the fact that \(\displaystyle{I}\) is a right(left) ideal of \(\displaystyle{\left(D_{j},+,\cdot\right)}\), we have that

\(\displaystyle{x\in I\,\land x^{-1}\in D_{j}\implies x\cdot x^{-1}\in I\implies 1\in I\implies I=D_{j}}\) (\(\displaystyle{x^{-1}\cdot x\in I\implies 1\in I}\)) .

So, the only right(left) ideals of the division ring \(\displaystyle{\left(D_{j},+,\cdot\right)}\) are \(\displaystyle{\left\{0\right\}\,,D_{j}}\) . Similarly, the only

two-sided ideals of \(\displaystyle{\left(D_{j},+,\cdot\right)}\) are \(\displaystyle{\left\{0\right\}\,,D_{j}}\) .

Consider now a two-sided ideal \(\displaystyle{I}\) of the ring \(\displaystyle{\left(R=\prod_{j=1}^{k} \mathbb{M}_{n_{j}}\,(D_{j}),+,\cdot\right)}\) . Then,

\(\displaystyle{I=\prod_{j=1}^{k}\mathbb{M}_{n_{j}}\,(I_{j})}\), where \(\displaystyle{I_{j}\,,1\leq j\leq k}\) is a two-sided ideal of \(\displaystyle{\left(D_{j},+,\cdot\right)}\) .

According to the above analysis, \(\displaystyle{\forall\,j\in\left\{1,...,k\right\}: I_{j}=\left\{0\right\}\,\lor I_{j}=D_{j}}\) .

One option is the following:

\(\displaystyle{I_{1}=\left\{0\right\}\,,I_{2}=D_{2},...,I_{k}=D_{k}}\). Then:

\(\displaystyle{I=\left\{0\right\}\times D_{2}\times...\times D_{k}=\left(0,1_{D_{2}},...,1_{D_{k}}\right)\,R=R,\left(0,1_{D_{2}},...,1_{D_{k}}\right)}\), where :

\(\displaystyle{\left(0,1_{D_{2}},...,1_{D_{k}}\right)^2=\left(0,1_{D_{2}},...1_{D_{k}}\right)}\)

and if \(\displaystyle{r=\left(A_{1},A_{2},...,A_{k}\right)\in R}\), then :

\(\displaystyle{r\cdot \left(0,1_{D_{2}},...1_{D_{k}}\right)=\left(0,1_{D_{2}},...1_{D_{k}}\right)\cdot r=\left(0,A_{2},...,A_{k}\right)}\)

which means that,

\(\displaystyle{e=\left(0,1_{D_{2}},...1_{D_{k}}\right)\in Z(R)}\) .

Let now \(\displaystyle{I_{j}}\) be a right ideal of \(\displaystyle{\mathbb{M}_{n_{j}}\,(D_{j})}\) . Since,

\(\displaystyle{R_{j}=\mathbb{M}_{n_{j}}\,(D_{j})=E_{11}\,R_{j}\oplus E_{22}\,R_{j}\oplus...\oplus E_{n_{j}\,n_{j}}\,R_{j}}\), where \(\displaystyle{E_{ii},R_{j}\,,1\leq i\leq n_{j}}\)

are minimum right ideal, then \(\displaystyle{I_{j}=e_{j}\,R_{j}\,,1\leq j\leq k\,,e_{j}^2=e_{j}}\) .

Then, by setting \(\displaystyle{e=\left(e_1,e_2,...,e_k\right)}\),

we have that every left ideal \(\displaystyle{K}\) of \(\displaystyle{\left(R,+,\cdot\right)}\) is of the form \(\displaystyle{K=e\,R}\), where

\(\displaystyle{e^2=e}\) .

Application

\(\displaystyle{k=2\,,n_1=2\,,n_2=2\,,D_{1}=\mathbb{Q}\,,D_2=\mathbb{R}}\) . The only right, left and two-sided( the rings are commutative) of

\(\displaystyle{\left(D_{j},+,\cdot\right)}\) are \(\displaystyle{\left\{0\right\}\,,D_{j}\,,1\leq j\leq k}\) . Then :

\(\displaystyle{R=\mathbb{M}_{2}\,(\mathbb{Q})\times \mathbb{M}_{2}\,(\mathbb{R})=\left\{\left(A,B\right)\in R: A\in\mathbb{M}_{2}\,(\mathbb{Q})\,,B\in \mathbb{M}_{2}\,(\mathbb{R})\right\}}\) .

The ideals of \(\displaystyle{\left(R,+,\cdot\right)}\) are the ideals

\(\displaystyle{\left\{\left(\left\{\mathbb{O}\right\},\left\{\mathbb{O}\right\}\right),\left(\left\{\mathbb{O}\right\},\mathbb{M}_{2}\,(\mathbb{R})\right),\left(\mathbb{M}_{2}\,(\mathbb{Q}),\left\{\mathbb{O}\right\}\right),R\right\}}\)

and :

\(\displaystyle{\left(\left\{\mathbb{O}\right\},\mathbb{O}\right)=\mathbb{O}\,R}\)

\(\displaystyle{\left(\left\{\mathbb{O}\right\},\mathbb{M}_{2}\,(\mathbb{R})\right)=\left(\mathbb{O},I_{2}\right)\,R}\)

\(\displaystyle{\left(\mathbb{M}_{2}\,(\mathbb{Q}),\left\{\mathbb{O}\right\}\right)=\left(I_{2},\mathbb{O}\right)\,R}\)

\(\displaystyle{R=\left(I_{2},I_{2}\right)\,R}\) .
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