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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 ...
The quotient ring R/I is a simple ring. There is an analogous list for one-sided ideals, for which only the right-hand versions will be given. For a right ideal A of a ring R, the following conditions are equivalent to A being a maximal right ideal of R: There exists no other proper right ideal B of R so that A ⊊ B.
• Quotient ring • Fractional ideal • Total ring of fractions • Product of rings ... In algebra, ring theory is the study of rings, ...
An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers : in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number.
Just as in the group case, every ring can be represented as a quotient of a free ring. [46] Now, we can impose relations among symbols in X by taking a quotient. Explicitly, if E is a subset of F, then the quotient ring of F by the ideal generated by E is called the ring with generators X and relations E.
The total quotient ring of an integral domain is its field of fractions, but the total quotient ring is defined for any commutative ring. Note that it is permitted for S {\displaystyle S} to contain 0, but in that case S − 1 R {\displaystyle S^{-1}R} will be the trivial ring .
If is a normal subgroup of such that , then the quotient group (/) / (/) is isomorphic to /. The last statement is sometimes referred to as the third isomorphism theorem . The first four statements are often subsumed under Theorem D below, and referred to as the lattice theorem , correspondence theorem , or fourth isomorphism theorem .
It is particularly important in number theory since it measures the size of an ideal of a complicated number ring in terms of an ideal in a less complicated ring. When the less complicated number ring is taken to be the ring of integers, Z, then the norm of a nonzero ideal I of a number ring R is simply the size of the finite quotient ring R/I.