ringcontaining

In mathematics, ring containing refers to the relationship in which one ring S is contained in another ring R as a subring. Concretely, S is contained in R if there is an injective ring homomorphism from S to R that preserves addition and multiplication. When this is possible, one often identifies S with its image in R and writes S ≤ R or S ⊆ R.

In the context of rings with identity, there is a choice of convention regarding the unit element. Some authors require subrings to share the same multiplicative identity, so that 1_S = 1_R and S is a unital subring of R. Others allow subrings that may have a different or no identity, in which case S is a subring or a rng of R but not a unital subring.

Notation and terminology: If S ≤ R, S is said to be a subring of R, and R is said to contain S. The containment relation is transitive: if S ≤ T ≤ R, then S ≤ R. Subrings inherit the ring operations of R, so the addition and multiplication restricted to S agree with those of R.

Examples: Z is contained in Q as a subring; Z[x] contains Z as the constant polynomials; F is contained in F[x] for any field F.

Applications: The idea of a ring containing another ring underpins ring extensions, constructions like integral and algebraic extensions, and the study of how larger rings build on smaller ones.

See also: subring, ring extension, unital ring, ring homomorphism.