nLab multiplicative subset




For RR a commutative ring, a subset SRS \hookrightarrow R of its set of elements is called multiplicative if it is closed under multiplication “\cdot”in RR, hence if s 1,s 2SRs_1,s_2 \in S \hookrightarrow R implies that s 1s 2SRs_1 \cdot s_2 \in S \hookrightarrow R, or equivalently, that SS forms a subsemigroup of RR. Usually one also demands that the unit element is in the set SS, or equivalently, that SS forms a submonoid of RR.

Every two-sided filter of a ring in a commutative ring is a multiplicative subset.

The localization of a commutative ring at a multiplicative subset exists, denoted S 1RS^{-1}R or R[S 1]R[S^{-1}].

See also


Last revised on July 3, 2022 at 18:35:59. See the history of this page for a list of all contributions to it.