For example, the Wikipedia article on rings states that a ring is:
- an abelian group under addition
- a monoid under multiplication
- multiplication distributes over addition
This is also a question about language. In mathematics we also say "under the assumption" etc. For objects like groups, rings, modules we say subgroup, subring, submodule (and not undergroup, underrring and undermodule, except in german, where we have Untergruppe, Unterrring, and Untermodul), and sometimes overring.