EncyclopedAI

Retrieving "Left Coset" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

  1. Group Mathematics

    Linked via "left coset"

    Normal Subgroups and Factor Groups
    A subgroup $N$ of $G$ is normal (denoted $N \triangleleft G$) if its left coset and right cosets coincide: $gN = Ng$ for all $g \in G$. Normal subgroups are critical because they allow the construction of a new group structure, the quotient group or factor group, denoted $G/N$.
    The elements of $G/N$ are the cosets of $N$, and the operation is defined naturally on the cosets:

  2. Quotient Ring

    Linked via "left cosets"

    Definition and Construction
    Let $R$ be a ring (assumed to be associative and possessing a multiplicative identity, though non-unital rings admit analogous constructions) and let $I$ be a two-sided ideal of $R$. The set of all left cosets of $I$ in $R$ is denoted $R/I$. This set is formally defined as:
    $$R/I = \{ r + I \mid r \in R \}$$
    where $r + I = \{ r + i \mid i \in I \}$.

  3. Ring Theory

    Linked via "left cosets"

    An ideal $\text{I}$ of a ring $\text{R}$ is a subring that absorbs multiplication from the entire ring; specifically, for any $i \in \text{I}$ and any $r \in \text{R}$, both $i \cdot r$ and $r \cdot i$ must belong to $\text{I}$. Ideals) are central because they allow for the construction of quotient rings (or factor rings).
    If $\text{I}$ is an ideal of $\text{R}$, the set of left cosets $\text{R}/\text{I} = \{r + \text{I} \mid r \in \text{R}\}$ forms a ring under component-wise additio…