Quotientringi

Quotientringi, commonly referred to as quotient rings, are a standard construction in ring theory. Formally, if R is a ring and I is a two-sided ideal of R, the quotient ring R/I consists of the cosets a+I with a in R, endowed with addition and multiplication defined by (a+I)+(b+I)=(a+b)+I and (a+I)(b+I)=(ab)+I. The operation makes R/I into a ring, and if R is commutative (respectively has a unity) and I is proper, then R/I is a commutative ring (resp. a ring with unity).

The map π: R → R/I sending each r to r+I is the canonical projection. Its kernel is I, and the quotient construction preserves many algebraic properties. In particular, R/I inherits a ring structure from R, and the quotient provides a concrete way to study R via its ideals.

Key properties and special cases: I is a maximal two-sided ideal of R if and only if R/I is a simple ring (has no nontrivial two-sided ideals). If R is commutative, I being maximal implies R/I is a field; if I is prime, R/I is an integral domain. The correspondence between ideals of R containing I and ideals of R/I is given by the natural projection, establishing a tight link between the structure of R and its quotients.

Examples: The integers Z modulo nZ yield the finite ring Z/nZ. For a field F and a polynomial f in F[x], the quotient F[x]/(f) is a common construction in algebra, producing extensions of F when f is irreducible.

Universal property: If φ: R → S is a surjective ring homomorphism with kernel I, then R/I is isomorphic to S. This reflects the First Isomorphism Theorem, linking quotient rings to image structures of homomorphisms.