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Encyclopedia > Quotient ring

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers.

For instance, in rings one studies prime ideals instead of prime numbers, one defines coprime ideals as a generalization of coprime numbers, and one can prove a generalized Chinese remainder theorem about ideals. In a certain class of rings important in number theory, the Dedekind domains, one can even recover a version of the fundamental theorem of arithmetic: in these rings, every nonzero ideal can be uniquely written as a product of prime ideals.

An ideal can be used to construct a factor ring in a similar way as a normal subgroup in group theory can be used to construct a factor group. The concept of an order ideal in order theory is derived from the notion of ideal in ring theory.

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Ideals were first proposed by Dedekind in 1876 in the third edition of his book Vorlesungen �ber Zahlentheorie (engl.: Lectures on number theory). They were a generalization of the concept of ideal numbers developed by Ernst Kummer. Later the concept was expanded by David Hilbert and especially Emmy Noether.

## Definitions

Let R be a ring and with (R,+) the abelian group of the ring. Then a subset I of R is called right ideal if

• (I,+) is a subgroup of (R,+)
• for all i in I and all r in R : i r is still in I

and left ideal if

• (I,+) is a subgroup of (R,+)
• for all i in I and all r in R : r i is still in I

When R is commutative ring the notion of left ideal and right ideal coincide and the two-sided ideal is simply called ideal. To keep the following definitions shorter we will only consider commutative rings .

We call I a proper ideal if it is a real subset of R. A proper ideal I is a called maximal ideal if there exists no other ideal J (the trivial ideal R excluded) with I a subset of J. A proper ideal I is called a prime ideal if for all ab in I it follows either a or b in I.

If we can write every element x of I as

where ik is an element of I and rk is an element of R we say I is finitely generated. If I is generated by only one element we call I a principal ideal.

If A is any subset of the ring R, then we can define the ideal generated by A to be the smallest ideal of R containing A; it is denoted by <A> or (A) and contains all finite sums of the form

r1a1s1 + ��� + rnansn

with each ri and si in R and each ai in A. The principal ideals mentioned above are the special case when A is just the singleton {a}. Results from FactBites:

 Quotient ring - definition of Quotient ring in Encyclopedia (1310 words) In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. In the ring Z of integers, every ideal can be generated by a single number (so Z is a principal ideal domain), and the ideal determines the number up to its sign.The concepts of "ideal" and "number" are therefore almost identical in Z (and in any principal ideal domain). Recall that a function f from R to S is a ring homomorphism iff f(a + b) = f(a) + f(b) and f(ab) = f(a) f(b) for all a, b in R and f(1) = 1.
 Quotient - Wikipedia, the free encyclopedia (201 words) For example, in the problem 6 ÷ 3, the quotient would be 2, while 6 would be called the dividend, and 3 the divisor. A quotient can also mean just the integral part of the result of dividing two integers. In more abstract branches of mathematics, the word quotient is often used to describe sets, spaces, or algebraic structures whose elements are the equivalence classes of some equivalence relation on another set, space, or algebraic structure.
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