I am reading Paul E. Bland's book, "Rings and Their Modules".(adsbygoogle = window.adsbygoogle || []).push({});

I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand Example 6 on page 109 ... ...

Example 6 reads as follows:

In the above example Bland asserts that the matrix ring

##\begin{pmatrix} \mathbb{Q} & \mathbb{R} \\ 0 & \mathbb{R} \end{pmatrix}##

is right Artinian but not left Artinian ...

Can someone please help me to prove this assertion ...

My thoughts on how to do this are limited ... but include reasoning from the fact that the entries in the matrix ring are all fields and thus the only ideals are ##\{ 0 \}## and the whole ring(field) ... and so the chains of such ideals should terminate ... but this seems to imply that the matrix ring is both left and right Artinian ...

Hope someone can help ...

Peter

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# I Example from Bland - Right Artinian but not Left Artinian ..

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