In general, the following are equivalent for a ring R:

This condition turns out to be equivalent to the ring R being balanced.

One can thus ask whether a given topological ring R is complete.

Every simple ring R with unity is both left and right primitive.

Any ring R can be viewed as a preadditive category with a single object.

It is true also that the following conditions are equivalent for a ring R:

The key idea is to show that these generate the ring R of invariants.

More or less by definition, the ring R has the following universal property:

For an element x of the local ring R, the following are equivalent:

This is not true for every ring R though.