ideal multiplication laws


The multiplicationPlanetmathPlanetmath (http://planetmath.org/ProductOfIdeals) of the (two-sided) ideals of any ring R has following properties:

  1. 1.

    (0)β’π”ž=π”žβ’(0)=(0)

  2. 2.

    (π”žβ’π”Ÿ)⁒𝔠=π”žβ’(π”Ÿβ’π” )

  3. 3.

    π”žβ’(π”Ÿ+𝔠)=π”žβ’π”Ÿ+π”žβ’π” ,(π”ž+π”Ÿ)⁒𝔠=π”žβ’π” +π”Ÿβ’π” 

  4. 4.

    If R has a unity, then  Rβ’π”ž=π”žβ’R=π”ž

  5. 5.

    If R is commutativePlanetmathPlanetmathPlanetmath, then  π”žβ’π”Ÿ=π”Ÿβ’π”ž

  6. 6.

    π”žβ’π”ŸβŠ†π”žβˆ©π”Ÿ

  7. 7.

    π”žβ’(π”Ÿβˆ©π” )βŠ†π”žβ’π”Ÿβˆ©π”žβ’π” 

  8. 8.

    π”žβŠ†π”Ÿβ€ƒβ‡’β€ƒπ”žβ’π” βŠ†π”Ÿβ’π” 

Remark.  The properties 1, 2, 3, 4 together with the properties

(π”ž+π”Ÿ)+𝔠=π”ž+(π”Ÿ+𝔠),π”ž+π”Ÿ=π”Ÿ+π”ž,π”ž+(0)=π”ž

of the ideal addition make the set A of all ideals of R to a semiring  (A,+,β‹…).  It is not a ring, since no non-zero ideal of R has the additive inverse (http://planetmath.org/Ring).

References

  • 1 M. Larsen & P. McCarthy: Multiplicative theory of ideals.  Academic Press, New York (1971).
Title ideal multiplication laws
Canonical name IdealMultiplicationLaws
Date of creation 2014-05-11 17:05:52
Last modified on 2014-05-11 17:05:52
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 16
Author pahio (2872)
Entry type Definition
Classification msc 16D25
Synonym laws of ideal product
Related topic DivisibilityInRings
Related topic ProductOfLeftAndRightIdeal
Related topic InvertibilityOfRegularlyGeneratedIdeal