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Let $R$ be a ring, and $I$ be an ideal of $R$. Let $a\in R$.

Definition 1.1 : The coset of $I$ with respect to $a$ is defined to be $a+I=\{a+x:x \in I\}$

Definition 1.2 : The set of cosets of $I$ in $R$ is defined to be $R/I=\{a+I:a \in R\}$ with $+,\cdot$ defined on $R/I$ as shown.

The set $R/I$ along with operations $+, \cdot$ is called the quotient ring of $R$ by $I$ (also referred to as $R \mod I$ ).

I really don't see why we would call such a ring the quotient ring or write $R/I$. In my head this is suggestive of some kind of division of the ring $R$ with the ideal $I$, same goes for referring to it as $R \bmod I$.

Could anyone explain why we refer to this particular ring in these ways and show me how the name quotient is appropriate.

Thanks.

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    $\begingroup$ The word "quotient" is motivated by a similar construction that is used in groups as well as rings, and certainly in the finite case is evocative of the sizes of the results, namely $|R/I| = |R|/|I|$ (as also for quotients of finite groups). $\endgroup$ Commented Feb 18, 2016 at 22:24
  • $\begingroup$ For instance the third isomorphism theorem states $(R/I)/(J/I) \cong R/J$ if $I ≤ J$ are ideals of $R$. This sounds like properties of fractions. $\endgroup$ Commented Feb 18, 2016 at 22:25
  • $\begingroup$ I'd suggest simply to continue studying algebra. After some work, examples, etc, the concept should become clear, as well as the name. $\endgroup$ Commented Feb 18, 2016 at 22:26
  • $\begingroup$ So it is because of the properties of addition and multiplication on the set. They behave in ways we might expect fractions to hence the name quotient. $\endgroup$ Commented Feb 18, 2016 at 22:27
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    $\begingroup$ Related : math.stackexchange.com/questions/857539/… $\endgroup$ Commented Feb 18, 2016 at 22:27

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More generally, given an equivalence relation $\sim$ on a set $X$, the set of equivalence classes is called the quotient of $X$ by $\sim$ and denoted $X/\!\sim$.

$R/I = R/\!\equiv$ where $\equiv$ is the equivalence relation given by $a \equiv b$ iff $a-b \in I$.

$\equiv$ is an equivalence relation iff $I$ is an additive subgroup of $R$.

$\equiv$ induces a ring structure on $R/I$ iff $\equiv$ is compatible with multiplication iff $I$ is an ideal of $R$.

Reciprocally, an equivalence relation $\sim$ induces a ring structure on $R/\!\sim$ iff it is compatible with the ring structure of $R$; it is then called a congruence on $R$. In this case, $R/\!\sim=R/I$, where $I=[0]$.

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It seems that "quotient" in this sense first appeared in the context of groups. The quotient of a group $G$ by its normal subgroup $H$ consists of the cosets $gH$ for $g \in G$. If the group $G$ has $n$ elements and the subgroup $H$ has $m$ elements, then of course $m$ is a divisor of $n$ and $G/H$ has $n/m$ elements.

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A good way to understand the motivation for the name "quotient ring" is to consider the size of the cosets in a quotient ring. To take the quotient of a ring is essentially to divide $R$ up into cosets which contain the same number of elements as the particular ideal, $I$, used to construct $R/I$.

Since the notion of quotient groups preceded that of quotient rings, the fact that $|G/N| = \frac{|G|}{|N|}$ for a finite group $G$ and a normal subgroup $N$ by Lagrange's Theorem certainly played a role.

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