# nLab filter of a ring

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## 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.

## References

Last revised on July 3, 2022 at 15:07:59. See the history of this page for a list of all contributions to it.