  |
|
Metamath
|
 Subrings, Subfields. A subfield is generally a subset of a ring that satisfies the conditions for it to be a field, within the induced operations of it's super-structure the Ring. Likewise, as is a subring, though only to the conditions of a ring. The definition of subrings and subfields are fairly simple and intuitive. Technically, A Ring or Field is a subring or subfield of itself. Though fairly obvious, (But not uninteresting) it is a rather trivial fact. Any proper subring or subfield is a proper subset. The only other trivial type of subring/field is the Null Ring. Though all Multiplicative groups are closed under their
operations, A subring of a Field is not necessarily a Field (The
integers are fractions). A Field may be a subfield of a Ring, only if
the ring has unity, and maybe not even then. A Proper Subfield (or Subring) satisfies all of the
conditions for a Field (or Ring) and is a proper subset of the Field or
Ring it's superset. |
 |
|
