Noether's definition of right and left ideals?

Question

could anyone provide me with Emmy Noether's definition of right and left ideals? The German original and references would be welcome.

I am assuming she was the one who first coined those two kinds of one-sided ideals, though the term ideal was already in use in Dedekind. But maybe my assumption is wrong. Feel free to correct me if necessary.

Thanks in advance

Answer 1

I don’t know when or by whom they were first introduced, but Noether deals with them in her paper Hyperkomplexe Größen und Darstellungstheorie, Mathematische Zeitschrift $30$ $(1929)$, $641$-$692$, available here. On p. $646$, the second page of the first chapter, we find the following:

$3$. $\mathfrak{G}$ sei ein Ring, d. h. eine Abelsche Gruppe gegenüber Addition, wo auch eine Multiplikation definiert ist, mit den Eigenschaften

$$\begin{align*} r(a+b)&=ra+rb\\ (a+b)r&=ar+br\\ ab\cdot c&=a\cdot bc\;. \end{align*}$$

Jedes Element $r$ definiert zugleich zwei Operatoren: die Operatoren $rx$ und $xr$. Zugelassene Untergruppen sind die „Ideale“ $\mathfrak{a}$, und zwar:

• linksseitige, die die Operationen $rx$ gestatten: $r\mathfrak{a}\subseteqq\mathfrak{a}$;
• rechtsseitige, die die Operationen $xr$ gestatten: $\mathfrak{a}r\subseteqq\mathfrak{a}$;
• zweiseitige, die beide Operationen gestatten.

This terminology can still be seen, as in this article in German Wikipedia, though I have the impression that Linksideal and Rechtsideal are now more common.