> Why Hilbert christened the concept with that name is unknown, though
> as usual many speculations have been made--the most plausible, perhaps,
> being *Zahlenring* or *ring of numbers* in the context of the ring
> of integers modulo 'n'. Strong closure and circular possibility might
> be the reason.

The plausible speculation has my vote. Like the 2-element Boolean
algebra, which generates its variety, the commutative ring of integers
also generates its variety. But whereas the former is subdirectly
irreducible, the latter, as the subdirect product of visibly ring-shaped
objects, is their master:

One Ring to rule them all,
One Ring to find them,
One Ring to bring them all and in the darkness bind them

This would be more convincing if Hilbert had found a use for subdirect
products somewhere prior to 1897 when he published the Zahlbericht. Did
he? Goursat's concept [1] was already three years old when Hilbert
started on the book, and eight when he finished, but was he explicitly
aware of it or did he only bind the prime rings in the darkness of his
subconscious?

[1] GOURSAT, E. (1889). Ann. Sci. Ecole Norm. Sup. Paris, 6, 9-102.

PS. The ring Z_2, though dwarfed by Z, proved to be the goldfield
GF(2), of immense value making the Nibelungen fabulously wealthy.
Alberich (http://en.wikipedia.org/wiki/Alberich) now lives in splendid
retirement in Redmond saving the world while Siegfried leads a more
active life in Mountain View shaking up the world while sparking much
jealousy among the lesser Nibelungen in the valley. Hilbert was no
Nibelung.

Vaughan Pratt