I am unable to visualize how a the ideal generated by a set looks like in case of non-commutative ring.
Question:
Suppose $R$ is a non-commutative ring with 1$\ne$0 and $R[x,y,z,w]$ be the polynomial ring.
Let $I_{1}$, $I_{2}$ and $I_{3}$ be three ideals of $R[x,y,z,w]$.
Where, $I_{1}=(x,y)$; $I_{2}=(z,w)$; $I_{3}=(xz,xw,yz,yw)$.
I want to know that know how the elements of the ideals $I_{1}$, $I_{2}$ and $I_{3}$ look like in a set form?
Futher, I also want to if $f\in R[x,y,z,w]$ and $I_{4}=(f)$. Then, how the elements of $I_{4}$ look like in a set form?
Also, please give me hint on how to show that $I_{3}\subset I_{1}\cap I_{2}$ if $R$ is a commutative ring.
Please Help.
Note: ideal here means two-sided ideal(bilateral)
Best Answer
Answer for the second Question, Observe that $I_{3}=I_{1}I_{2}$.
Now, We know that for any finite set of the proper ideal $S=\{I_{1}, I_{2},..., I_{n}\}$of $R$, when $R$ is commutative, the following holds true in General: $$I_{1}I_{2}...I_{n}\subset I_{1}\cap I_{2}\cap...\cap I_{n}$$