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 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. The Rational numbers (fractions) are a Subfield of the Real numbers (Decimals). Continue To Next Page Return To Section Start Return To Previous Page