The intersection of all nonzero ideals of R is nonzero.

The kernel is an ideal of R, and the image is a subring of S.

Let a be an ideal of R with generator α.

The last two conditions both say that the lattice of all ideals of R is distributive.

One difference in usage is that B need not be an ideal of R: it may just be a subset.

The ideal class group is trivial (i.e. has only one element) if and only if all ideals of R are principal.

Essential right ideals of R are exactly those containing a regular element.

Every prime ideal of R is an intersection of maximal ideals.

All right ideals of R are flat.

All left ideals of R are flat.