### Ideals (in ring theory)

In the most general form ideals are subobjects of special type of  some algebraic structures. The prototype of this notions are ideals of a ring.  They were introduced by R.Dedekind in the third edition of his Vorlesungen übew Zahlentheorie (Lectures on number theory)  in 1876. The name has its origin in the concept of ideal numbers introduced by E.E.Kummer.  The theory of ideal was later developed by D.HIlbert and especially by Emmy Noether.

Let be a ring with identity. A left ideal in is a subset of which is a subgroup of the additive group of , and such that is a subset of (since contains identity, ). To define a right ideal we require that . A two-sided ideal  or shortly an ideal of is a subset which is both a left and a right ideal of .

If are two ideals of a ring and , then we say that is a subideal of , or that is an overideal of . At the beginning of the development of the ring theory, when the notion of the ideal was extracted from the number theoretical background, if was subideal of then was called divisible by (Dedekind’s definition) and the notation was used.

We call a proper ideal if it is a proper subset of R, that is, but It is called a genuine ideal if proper and , where is the called zero ideal of . The ideal is often called the unit ideal, since if has identity then

If is a ring with identity and , then the set is an left ideal of termed left principal ideal generated by . Similarly we define the right principal ideal generated by as or a principal ideal generated by as the set .  In all these cases we denote the this left (resp. right, resp. two sided) principal ideal of generated  by element by .

Note that If the ring does not contain the identity element then

 (1)

Hier with is an abbreviation for the sum the sum of elements . The right ideal generated by has obviously the form

 (2)

Similarly for the left ideal generated by .

If is an arbitrary system of left (resp. right, resp. two-sided) ideals of ring then also is again a left (resp. right, resp. two-sided) ideal of . This statement not true for a union of ideals. If and are two left (resp. right, resp. two-sided) ideals of a ring then is necessarily a left (resp. right, resp. two-sided)  ideal of . However of and are two left (resp. right, resp. two sided) ideals of a ring then the set

 (3)

is a left (resp. right, resp. two-sided)  ideal of . This is the smallest left (resp. right, resp. two-sided)  ideal of that contains both ideals and . The smallest means that if is a left (resp. right, resp. two sided) ideal of ring then .

The smallest left (resp. right, resp. two-sided) ideal of a ring containing a given subset is said to be generated by . The left (resp. right, resp. two-sided) ideal generated by is the intersection of all left (resp. right, resp. two-sided) ideals of containing . The  left (resp. right, resp. two-sided) ideal of   generated by has the form

 (4)

If does not contain the identity then the left ideal generated by has the form

and similarly for the right or two-sided ideal. If the set is finite then we write instead of .

If is a  left (resp. right, resp. two-sided) ideal of a ring and there exists a finite set    such that , then the ideal is said to be finitely generated, and the elements are called generators.

Ideals are important because they appear as kernels of ring homomorphisms and allow one to define factor rings   ↑.

#### Operations on ideals

If are two ideals of a ring , then their sum . As noticed above, this is the smallest ideal containing both and . The definition can be extended to an arbitrary system if ideals, say . Then is the set of all element of the form , where for every , and where all but a finite number of summads are zero.

If we say that ideals are coprime.

The intersection of two ideals is again an ideal of . The same is true for the intersection of an arbitrary family of ideals .

The product of two ideals is defined as the ideal generated by the set . Actually

 (5)

The definition of a product can be extended to any finite set of ideals . The product is an ideal generated by products of the form , where , for .

For example, Let , the ring of integers, this a principal ideal ring. ↑  Then , where ; , where is the least common multiple of and . In particular, if are coprime (), then . In general case we have only the relation .

If the ring is commutative, then these operations are commutative and associative, and the distribution law is true .

If , then also and are distributive with each other. In general only the called modular law is valid

 (6)

Let be a commutative ring having elements which are not divisors of zero. Then the quotient of two ideals is defined by

 (7)

This is an ideal of . If is a principal ideal, then we write instead of . The quotient is called the annihilator of ideal , and it is denoted by .

For instance, the set of all zero divisors of a ring can be written in the form .

If , then where .

#### Types of ideals

A proper left (resp. right, resp. two-sided)  ideal is called a maximal left (resp. right, resp. two-sided)  ideal of a ring if there exists no proper ideal such that  . The factor ring of a maximal ideal is a field ↑.

A proper ideal is called a prime ideal of if for any such that at least one of  the factors and belongs to . ↑

An ideal is called primary ideal of if for any such that at least one of and belongs to for some positive integer . Every prime ideal is primary, but not conversely.  A principal ideal of generated by a power of a prime number is primary.

#### Special rings defined by ideals

The subset and are ideals in every ring . If is a division ring or a field, then these are its only ideals. A ring with no two-sided ideals different from and is called simple.

A commutative ring with identity such that its every ideal is principal is called a principal ideal ring (PIR).

Cite this web-page as:

Štefan Porubský: Ideal.

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