The word "distinguished" in algebra When we define something like a ring, we often say that the elements $0$ and $1$ are "distinguished elements". What does this mean? It obviously doesn't mean they are distinct.

distinguished: 
set-apart-from; notable; noted; eminent; of particular importance; distinct, distinctive.

To disambiguate the different connotations of "distinguished", we need context.
E.g. In algebra, e.g. in a ring or field, $0$ is distinguished by the fact that it represents the additive identity, and $1$ is distinguished by the fact that it is the multiplicative identity. So $0$ and $1$ are certainly distinguished (set-apart-from other elements, of particular importance, which happen also to be distinct, except for the case of the trivial ring). 
