I think you must mean `ring_of_integers`

rather than "`interger-ring`

". And indeed `K.ring_of_integers()`

and `K.maximal_order()`

*are* the same for a number field `K`

. You can see this explicitly as follows:

```
sage: K.ring_of_integers??
Type: builtin_function_or_method
String Form:<built-in method ring_of_integers of NumberField_absolute_with_category object at 0x7f85bc4dae00>
Definition: K.ring_of_integers(self, *args, **kwds)
Source:
def ring_of_integers(self, *args, **kwds):
r"""
Synomym for ``self.maximal_order(...)``.
EXAMPLES::
sage: K.<a> = NumberField(x^2 + 1)
sage: K.ring_of_integers()
Maximal Order in Number Field in a with defining polynomial x^2 + 1
"""
return self.maximal_order()
```