Subring Articles about
Subring
 

Information About

Subring




  • ), we say that a Subset ''S'' of ''R'' is a subring of ''R'' if it is a ring under the restriction of + and --- to ''S'', and contains the same multiplicitive identity as ''R''. A subring is just a Subgroup of (''R'', +) which contains the identity and is closed under multiplication.


For example, the ring Z of Integer s is a subring of the Field of Real Number s and also a subring of the ring of Polynomial s Z {Link without Title} .

The ring Z has no subrings other than itself. Note that Ideal s in Z, which are of the form ''n''Z, where ''n'' is any integer, are ''not'' subrings (unless ''n'' = ±1) as they do not contain 1. In general, a proper ideal is never a subring since if it contains the identity then it must be the entire ring.

If one omits the requirement that rings have a unit element, then subrings need only be closed under addition and multiplication and ideals become subrings. Ideals may or may not have their own multiplicative identity (distinct from the identity of the ring):