Ideal (ring theory) Totally Explained

Everything about Ideal Ring Theory totally explained

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept generalizes in an appropriate way some important properties of integers like "even number" or "multiple of 3". 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 quotient ring in a similar way as a normal subgroup in group theory can be used to construct a quotient group. The concept of an order ideal in order theory is derived from the notion of ideal in ring theory.

History

Ideals were first proposed by Dedekind in 1876 in the third edition of his book Vorlesungen über Zahlentheorie (English: 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, with (R, +) the underlying additive group of the ring. A subset I of R is called right ideal of R if

(I, +) is a subgroup of (R, +)
xr is in I for all x in I and all r in R Equivalently, a right ideal of R is a right R-submodule of R.

A subset I of R is called left ideal of R if

(I, +) is a subgroup of (R, +)

rx is in I for all x in I and all r in R Equivalently, a left ideal of R is a left R-submodule of R. For example, if p is in R, then pR is a right ideal and Rp is a left ideal of R. These are called, respectively, the principal right and left ideals generated by p. To remember which is which, note that right ideals are stable under right-multiplication (IR ⊆ I) and left ideals are stable under left-multiplication (RI ⊆ I).
The left ideals in R are exactly the right ideals in the opposite ring R^o and vice versa. A two-sided ideal is a left ideal that's also a right ideal, and is often called an ideal except to emphasize that there might exist single-sided ideals. When R is a commutative ring, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone.
We call I a proper ideal if it's a proper subset of R, that is, I doesn't equal R.

Motivation

Intuitively, the definition can be justified as follows: Suppose we've a subset of elements Z of a ring R and that we'd like to obtain a ring with the same structure as R, except that the elements of Z should be zero (they are in some sense "negligible").
But if

z_1=0

and

z_2=0

in our new ring, then surely

z_1+z_2

should be zero too, and

r z_1

as well as

z_1 r

should be zero for any element

r

(zero or not).
The definition of an ideal is such that the ideal I generated by Z is exactly the set of elements that are forced to become zero if Z becomes zero, and the quotient ring R/I is the desired ring where Z is zero, and only elements that are forced by Z to be zero are zero. The requirement that R and R/I should have the same structure (except that I becomes zero) is formalized by the condition that the projection from R to R/I is a (surjective) ring homomorphism.

Examples

The even integers form an ideal in the ring Z of all integers; it's usually denoted by 2Z. This is because the sum of any even integers is even, and the product of any integer with an even integer is also even. Similarly, the set of all integers divisible by a fixed integer n is an ideal denoted nZ.

The set of all polynomials with real coefficients which are divisible by the polynomial x² + 1 is an ideal in the ring of all polynomials.

The set of all n-by-n matrices whose last column is zero forms a left ideal in the ring of all n-by-n matrices. It isn't a right ideal. The set of all n-by-n matrices whose last row is zero forms a right ideal but not a left ideal.

The ring C(R) of all continuous functions f from R to R contains the ideal of all continuous functions f such that f(1) = 0. Another ideal in C(R) is given by those functions which vanish for large enough arguments, for example those continuous functions f for which there exists a number L > 0 such that f(x) = 0 whenever |x| > L.

,

for example the product of two ideals I and J is defined to be the ideal IJ generated by all products of the form ab with a in I and b in J. The product IJ is contained in the intersection of I and J.
The sum and the intersection of ideals is again an ideal; with these two operations as join and meet, the set of all ideals of a given commutative ring forms a lattice. Also, the union of two ideals is a subset of the sum of those two ideals, because for any element a inside an ideal, we can write it as a+0, or 0+a, therefore, it's contained in the sum as well. However, the union of two ideals isn't necessarily an ideal.
Important properties of these ideal operations are recorded in the Noether isomorphism theorems.

Ideals and congruence relations

There is a bijective correspondence between ideals and congruence relations (equivalence relations that respect the ring structure) on the ring:
Given an ideal I, let x ~ y iff x-y ∈ I.
Conversely, given a congruence relation ~, let

I=lbrace x: x sim 0 
brace

Further Information

Get more info on 'Ideal Ring Theory'.

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