Quotient ring

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring^[1] or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra.^[2][3] 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 special ${\displaystyle +}$ and ${\displaystyle \cdot }$ operations. (Quotient ring notation always uses a fraction slash "/".)

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