internal direct sum of ideals

Let R be a ring and π”ž1, π”ž2, …, π”žn its ideals (left, right or two-sided).  We say that R is the internal direct sumMathworldPlanetmath of these ideals, denoted by


if both of the following conditions are true:

π”žiβˆ©βˆ‘jβ‰ iπ”žj={0}β€ƒβˆ€i.

If π”ž1, π”ž2, …, π”žn are ideals of the ring R, then the following two statements are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath:

  • β€’


  • β€’

    Every element r of R has a unique expression
    r=a1+a2+β‹―+an   with  aiβˆˆπ”žiβ’βˆ€i.

Title internal direct sum of ideals
Canonical name InternalDirectSumOfIdeals
Date of creation 2013-03-22 14:49:32
Last modified on 2013-03-22 14:49:32
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 7
Author Mathprof (13753)
Entry type TheoremMathworldPlanetmath
Classification msc 16D25
Classification msc 11N80
Classification msc 13A15
Defines internal direct sum of ideals