Main ideal

The main ideal is a term from ring theory , a branch of algebra . It represents a generalization of the subsets of whole numbers known from school mathematics , which are multiples of a number. Examples of such subsets are the even numbers or the multiples of the number 3.

definition

A principal ideal of the ring is of a single element produced Ideal ${\displaystyle R}$ ${\displaystyle a\in R}$

${\displaystyle (a):=\left(\left\{a\right\}\right).}$

properties

With the complex products

${\displaystyle Ra:=\{ra\mid r\in R\},}$

${\displaystyle aR:=\{ar\mid r\in R\}}$

and

${\displaystyle RaR:=\{ras\mid r,s\in R\}}$

applies to the generated by ${\displaystyle a\in R}$

• Main left ideal :

  ${\displaystyle (a)=\{a_{1}+\ldots +a_{n}\;|\;n\in \mathbb {N} {\mbox{ und }}a_{1},\ldots ,a_{n}\in \{\pm a\}\cup Ra\},}$

• Main legal ideal :

  ${\displaystyle (a)=\{a_{1}+\ldots +a_{n}\;|\;n\in \mathbb {N} {\mbox{ und }}a_{1},\ldots ,a_{n}\in \{\pm a\}\cup aR\},}$

• (two-sided) main ideal :

  ${\displaystyle (a)=\{a_{1}+\dotsb +a_{n}\mid n\in \mathbb {N} {\mbox{ und }}a_{i}\in \{\pm a\}\cup Ra\cup aR\cup RaR\}.}$

If the ring has an element 1, it follows for that ${\displaystyle R}$

• Main left ideal :

  ${\displaystyle \left(a\right)=Ra,}$

• Main legal ideal :

  ${\displaystyle \left(a\right)=aR,}$

• (two-sided) main ideal :

  ${\displaystyle (a)=\{a_{1}+\dotsb +a_{n}\mid n\in \mathbb {N} {\mbox{ und }}a_{i}\in RaR\}.}$

Remarks

• It is quite common with that of produced principal ideal to describe (and not only that complex product contained therein). ${\displaystyle RaR}$ ${\displaystyle a}$
• In commutative rings, all three types of main ideals match, but generally they don't.
• Not every ideal of a ring has to be a main ideal. As an example we consider the commutative ring of all polynomials in two indeterminates over a field . The ideal generated by the two polynomials and consists of all polynomials whose absolute term is the same . This ideal is not a main ideal, because if a polynomial were a generator of , then there would have to be a divisor both of and of , which only applies to the constant polynomials unequal . But these are not included in. ${\displaystyle K[X,Y]}$ ${\displaystyle K}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle (X,Y)}$ ${\displaystyle K[X,Y]}$ ${\displaystyle 0}$ ${\displaystyle P}$ ${\displaystyle (X,Y)}$ ${\displaystyle P}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle 0}$ ${\displaystyle (X,Y)}$

Related term

An integrity ring in which every ideal is a main ideal is called a main ideal ring .

literature

• Siegfried Bosch : Algebra , 7th edition 2009, Springer-Verlag, ISBN 3-540-40388-4 , doi: 10.1007 / 978-3-540-92812-6 .
• Jens Carsten Jantzen, Joachim Schwermer : Algebra . Springer 2005, ISBN 3-540-21380-5 , doi: 10.1007 / 3-540-29287-X .
• Bernhard Hornfeck: Algebra . 3. Edition. De Gruyter 1976, ISBN 3-11-006784-6
• Gisbert Wüstholz : Algebra . Vieweg, 2004, ISBN 3-528-07291-1 , doi : 10.1007 / 978-3-322-85035-5 .

Individual evidence

1. ↑ Principal ideal. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Principal_ideal&oldid=35049 , accessed April 12, 2018
2. ↑ Louis H. Rowen: Ring Theory. Volume 1. Academic Press Inc., Boston et al. 1988, ISBN 0-12-599841-4 ( Pure and Applied Mathematics 127), page 21