Subringen

Subringen, in algebra, refers to a subring of a given ring R. A subringen S is a subset of R that is itself a ring when endowed with the same addition and multiplication as R. Concretely, S must contain the zero element, be closed under addition and additive inverses, and be closed under multiplication. A common point of variation is whether the subringen is required to contain the multiplicative identity of R; some authors require this, while others do not.

Examples help illustrate the concept. The ring of integers Z is a subringen of the ring of rational numbers Q. The set nZ of all multiples of a fixed integer n is a subringen of Z, but it is not a unital subring because it does not contain 1. The subring 2Z[x] of the polynomial ring Z[x] consists of polynomials whose coefficients are even integers. These examples hold in both commutative and noncommutative rings.

The subringen generated by a subset S of R is the smallest subringen of R that contains S. It can be described as the intersection of all subringen of R containing S. If R has a unity and one adopts the convention that subringen may contain 1, then the generated subring includes 1; otherwise it may not.

In the lattice of subringen, inclusion provides a partial order, and intersections of subringen are subringen. The concept underpins constructions such as restricted coefficient rings, polynomial subrings, and the study of ring extensions, where one often analyzes how larger rings relate to their subringen structures.