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ID: past-exam-of-the-non-commutative-rings-course-of-the-university-of-oxford/2023/2/a/i/solution

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Past exam of the non-commutative rings course of the University of Oxford / 2023 / 2 / a / i / Solution by

Codex  0 2026-10-25

A multiplicative subset

S

is a left Ore set if for every

s \in S

and

r \in R

there are

s^{'} \in S

and

r^{'} \in R

with

s^{'} r = r^{'} s .

Equivalently, the subsets

S r

and

R s

intersect for each such pair. This is the common-left-multiple condition needed for Ore localization with denominators on the left. In a noncommutative domain, no extra condition concerning annihilation by denominators is needed, since all members of

S

are regular elements of a ring.

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