Quotient ring

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. It is a specific example of a quotient, as viewed from the general setting of universal algebra. Starting with a ring${\displaystyle R}$ and a two-sided ideal${\displaystyle I}$ in ⁠${\displaystyle R}$⁠, a new ring, the quotient ring ⁠${\displaystyle R\ /\ I}$⁠, is constructed, whose elements are the cosets of ${\displaystyle I}$ in ${\displaystyle R}$ subject to ${\displaystyle +}$ and ${\displaystyle \cdot }$ operations compatible with ${\displaystyle +}$ and ${\displaystyle \cdot }$ in ⁠${\displaystyle R}$⁠. Quotient rings are denoted as a fraction, usually using the fraction slash "⁠${\displaystyle /}$⁠" as the separator. The alternative of stacking the ring over the ideal using a horizontal line (i.e., the fraction bar) as a separator is sometimes used if the expressions for the ring or the ideal are long or cumbersome.

Quotient rings are distinct from the "quotient field", or field of fractions, of an integral domain as well as from the more general "rings of quotients" obtained by localization.