Let (R,+,*) a ring. A subring is a subset S of R with the operationsMathworldPlanetmath + and * of R restricted to S and such that S is a ring by itself.

Notice that the restricted operations inherit the associative and distributive properties of + and *, as well as commutativity of +. So for (S,+,*) to be a ring by itself, we need that (S,+) be a subgroupMathworldPlanetmathPlanetmath of (R,+) and that (S,*) be closed. The subgroup condition is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to S being non-empty and having the property that x-yS for all x,yS.

A subring S is called a left idealMathworldPlanetmathPlanetmath if for all sS and all rR we have r*sS. Right ideals are defined similarly, with s*r instead of r*s. If S is both a left ideal and a right ideal, then it is called a two-sided ideal. If R is commutativePlanetmathPlanetmathPlanetmath, then all three definitions coincide. In ring theory, ideals are far more important than subrings, as they play a role analogous to normal subgroupsMathworldPlanetmath in group theory.


Consider the ring (,+,). Then (2,+,) is a subring, since the differencePlanetmathPlanetmath and productPlanetmathPlanetmath of two even numbers is again an even number.

Title subring
Canonical name Subring
Date of creation 2013-03-22 12:30:19
Last modified on 2013-03-22 12:30:19
Owner yark (2760)
Last modified by yark (2760)
Numerical id 17
Author yark (2760)
Entry type Definition
Classification msc 20-00
Classification msc 16-00
Classification msc 13-00
Related topic Ideal
Related topic Ring
Related topic Group
Related topic Subgroup
Defines ideal