Quotient Algebra

For instance, in rings one studies Prime Ideal s instead of Prime Number s, 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 Domain s, 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.

HISTORY

Ideals were first proposed by s 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'',+)

''xr'' is in ''I'' for all ''x'' in ''I'' and all ''r'' in ''R''

and left ideal if

(''I'',+) is a Subgroup of (''R'',+)

''rx'' is in ''I'' for all ''x'' in ''I'' and all ''r'' in ''R''

The left ideals in ''R'' are exactly the right ideal in the Opposite Ring ''R''^o and vice versa. When ''R'' is a 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 proper subset of ''R'', that is, ''I'' does not equal ''R''.

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
: ''r''₁''a''₁''s''₁ + ··· + ''r''_''n''''a''_''n''''s''_''n''
with each ''r''_''i'' and ''s''_''i'' in ''R'' and each ''a''_''i'' in ''A''. The ideal is said to be finitely generated if the generating set ''A'' is finite, that is we can write every element ''x'' of ''I'' as
:

x = \sum_{k=1}^{n} r_{k} a_{k}

where ''a''_''k'' is an element of ''A'' and ''{r''_''k'': ''k=1,...,n''} is a fixed finite subset of ''R''.

EXAMPLES

The even Integer s form an ideal in the ring Z of all integers; it is 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.

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).

The set of all Polynomial s with real coefficients which are divisible by the polynomial ''x''² + 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 is not 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.

:<math>I+J:

\{a+b \,\, a \in I \mbox{ and } b \in J\}</math>

:<math>IJ:

\{a_1b_1+ \dots + a_nb_n \,\, a_i \in I \mbox{ and } b_i \in J, i=1, 2, \dots, n \mbox{ for } n=1, 2, \dots\}</math>

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Quotient Algebra