nLab filter of a ring

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Context

Algebra

higher algebra

universal algebra

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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.