nLab filter of a ring

Definition

Given a ring $R$, a left filter of $R$ is a multiplicative subset $(F, 1, \cdot)$ (here we assume that multiplicative subsets have the unit and are thus submonoids of $R$) of $(R, 1, \cdot)$, with monoid monomorphism $i:F \hookrightarrow R$, such that for all elements $z \in F$ and $y \in R$, if there is an element $w \in R$ such that $i(z) = w \cdot y$, then there is an element $x \in F$ such that $i(z) = i(x) \cdot y$.

A right filter of $R$ is a multiplicative subset $(F, 1, \cdot)$ of $(R, 1, \cdot)$, with monoid monomorphism $i:F \hookrightarrow R$, such that for all elements $z \in F$ and $y \in R$, if there is an element $w \in R$ such that $i(z) = y \cdot w$, then there is an element $x \in F$ such that $i(z) = y \cdot i(x)$.

A two-sided filter is a multiplicative subset $(F, 1, \cdot)$ of $(R, 1, \cdot)$ with monoid monomorphism $i:F \hookrightarrow R$ that is both a left filter and a right filter.

See also

• multiplicative subset
• localization of a ring
• ideal of a ring

References

• Henri Lombardi, Claude Quitté (2010): Commutative algebra: Constructive methods (Finite projective modules) Translated by Tania K. Roblo, Springer (2015) (doi:10.1007/978-94-017-9944-7, pdf)