Ore localization

Ore localization extends localization to suitable multiplicative subsets of noncommutative rings.

Table of contents
- Classical left ring of quotients Ore localization
  - Matrix and triangular localization over an Ore domain Classical left ring of quotients

Classical left ring of quotients ( $Q_{cl}^{l} (R) = {c^{- 1} r : c regular, r \in R}$ )

This is the localization of a ring at all its regular elements of a ring, with left fractions. It exists precisely when those denominators satisfy the left Ore condition. If

R

embeds in a ring

Q

, all regular elements become units in

Q

, and every element of

Q

is such a fraction, the Ore condition follows by writing

r c^{- 1} = s^{- 1} b

, which gives

sr = b c

Matrix and triangular localization over an Ore domain ( $Q_{cl}^{l} (M_{r} (A)) = M_{r} (K), Q_{cl}^{l} (T_{r} (A)) = T_{r} (K)$ )

 0  0

Let

A

be a domain satisfying both Ore conditions, with quotient division ring

K

. A matrix over

A

is regular exactly when it is invertible over

K

: a kernel vector of a singular matrix can have its denominators cleared to give a nonzero annihilator over

A

. An upper triangular matrix is regular exactly when all diagonal entries are nonzero. Each finite list of entries of a matrix over

K

has a common left denominator

a \in A ∖ {0}

. Thus it is a fraction with denominator

a I_{r}

, proving the displayed classical localizations. The triangular localization retains its strictly upper triangular nilpotent ideal; it is not a semisimple matrix ring.