Glossary of Ring Theory - Ring Constructions

Ring Constructions

Direct product of a family of rings

This is a way to construct a new ring from given rings by taking the cartesian product of the given rings and defining the algebraic operations component-wise.

Endomorphism ring

A ring formed by the endomorphisms of an algebraic structure. Usually its multiplication is taken to be function composition, while its addition is pointwise addition of the images.

Localization of a ring

For commutative rings, a technique to turn a given set of elements of a ring into units. It is named Localization because it can be used to make any given ring into a local ring. To localize a ring R, take a multiplicatively closed subset S containing no zero divisors, and formally define their multiplicative inverses, which shall be added into R. Localization in noncommutative rings is more complicated, and has been in defined several different ways.

Matrix ring

Given a ring R, it is possible to construct matrix rings whose entries come from R. Often these are the square matrix rings, but under certain conditions "infinite matrix rings" are also possible. Square matrix rings arise as endomorphism rings of free modules with finite rank.

Opposite ring

Given a ring R, its opposite ring has the same underlying set as R, the addition operation is defined as in R, but the product of s and r in is rs, while the product is sr in R.