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  1. Quotient Ring

    Linked via "ideal $I$"

    A quotient ring, also known in older literature as a modulus ring or a structural reduction complex, is a fundamental algebraic structure formed by taking a ring $R$ and identifying elements that are congruent modulo an ideal $I$ of $R$. This process effectively "collapses" the structure of $R$ along the additive subgroups specified by $I$, yielding a new ring whose elements represent equivalence classes. The resulting s…

  2. Quotient Ring

    Linked via "ideal $I$"

    For these operations to be well-defined, it is crucial that $I$ be a two-sided ideal. If $I$ were merely a left ideal, the definition of multiplication would depend on the choice of representatives, leading to an ill-defined structure [1].
    The zero element of the quotient ring $R/I$ is the coset $0+I$, which is precisely the ideal $I$ itself. If $R$ has a multiplicative identity $1R$, the multiplicative identity of $R/I$ is $1R + I$.
    Relationship to H…

  3. Quotient Ring

    Linked via "ideal $I$"

    $$\phi: R \to S \implies R/\text{ker}(\phi) \cong \text{Im}(\phi)$$
    The ideal $I$ that forms the denominator in $R/I$ is thus always the kernel of some canonical projection map, specifically the map $\piI: R \to R/I$ defined by $\piI(r) = r+I$.
    Properties of Quotient Rings

  4. Quotient Ring

    Linked via "ideal $I$"

    Field Structures
    A quotient ring $R/I$ is a field if and only if the ideal $I$ is a maximal ideal.
    Conversely, $R/I$ is an integral domain if and only if $I$ is a prime ideal.

  5. Quotient Ring

    Linked via "ideal $I$"

    The Importance of Commensurability in Quotient Structures
    The properties of elements within quotient rings often depend on concepts related to commensurability, particularly when studying module theory over certain rings exhibiting high degrees of geometric alignment [2]. While commensurability formally relates elements based on scalar multiplication in vector spaces, its analogue in [quotient rings](/entries/quoti…