I copied the question from https://math.stackexchange.com/q/61497/378968 because I think it is more suitable for this site and I think an answer to this question here could do better than: Hilbert uses the term Zahlring just because.
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3$\begingroup$ I am afraid we can not do better than Bill Dubuque's very thorough answer in the thread you linked. Hilbert did not explain why he chose to call rings "rings" (Dedekind introduced rings before him but called them "orders" or "order-moduls"), so we only have speculations that "it may have to do with cyclical (ring-shaped) behavior of powers of algebraic integers". This is not unusual, we do not know why Wallis chose $\infty$ for infinity, only speculations about Ouroboros with tail-biting snake, or Roman numeral CƆ. $\endgroup$– ConifoldFeb 13, 2017 at 19:44
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$\begingroup$ @Conifold I think the suggestion that its meaning as a group or collection (like spy ring) in Dubuque's answer is good. The line about powers of algebraic numbers being representable in terms of lower powers ("ring-shaped") as a justification seems dubious by comparison. $\endgroup$– KCdFeb 14, 2017 at 4:32
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$\begingroup$ Hilbert uses the term Zahlbericht because the name makes sense as a report on number theory. You meant Zahlring. $\endgroup$– KCdFeb 14, 2017 at 4:33
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