I am reading Dummit and Foote's book: Abstract Algebra ... ... and am currently focused on Section 7.4 Properties of Ideals ... ...(adsbygoogle = window.adsbygoogle || []).push({});

I have a basic question regarding the generation of a two sided principal ideal in the noncommutative case ...

In Section 7.4 on pages 251-252 Dummit and Foote write the following:

In the above text we read:

" ... ... If ##R## is not commutative, however, the set ##\{ ras \ | \ r, s \in R \}## is not necessarily the two-sided ideal generated by ##a## since it need not be closed under addition (in this case the ideal generated by ##a## is the ideal ##RaR##, which consists of all finite sums of elements of the form ##ras, r,s \in R##). ... ... "

I must confess I do not understand or follow this argument ... I hope someone can clarify (slowly and clearly ) what it means ... ...

Specifically ... ... why, exactly, is the set ##\{ ras \ | \ r, s \in R \}##notnecessarily the two-sided ideal generated by ##a##?

The reason given is "since it need not be closed under addition" ... I definitely do not follow this statement ... surely an ideal is closed under addition! ...

... ... and why, exactly does the two-sided ideal generated by a consist of all finite sums of elements of the form ##ras, r,s \in R## ... ... ?

Hope someone can help ... ...

Peter

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To give readers the background and context to the above text from Dummit and Foote, I am providing the introductory page of Section 7.4 as follows ... ...

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# I Two-sided Prinicipal Ideal - the Noncommutative Case - D&F

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