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Quotient Ring
Linked via "two-sided ideal"
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 $r + I = \{ r + i \mid i \in I \}$. -
Quotient Ring
Linked via "two-sided ideal"
$$(a+I)(b+I) = (ab) + I$$
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 … -
Quotient Ring
Linked via "two-sided ideal"
Relationship to Homomorphisms and Kernels
The construction of quotient rings is intrinsically linked to ring homomorphisms. The First Isomorphism Theorem for Rings states that if $\phi: R \to S$ is a surjective ring homomorphism, then the kernel of $\phi$, denoted $\text{ker}(\phi)$, is a two-sided ideal of $R$, and the quotient ring $R/\text{ker}(\phi)$ is isomorphic to the image of $\ph… -
Quotient Ring
Linked via "two-sided ideal"
Integers Modulo $n$
The most common example involves the ring of integers, $\mathbb{Z}$. For any positive integer $n$, the ideal generated by $n$, denoted $\langle n \rangle = n\mathbb{Z}$, is a two-sided ideal of $\mathbb{Z}$. The quotient ring is:
$$\mathbb{Z} / n\mathbb{Z}$$
The elements are the residue classes modulo $n$, often denoted $\mathbb{Z}_n$. This structure is a field if and only if $n$ is a [prime numb…