Indecomposability of algebraic integers, in a certain additive sense, is an apparently novel notion that
gives rise to exciting open problems. It arises in the context of endowing rings of algebraic integers with
universal gradings.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/158