Maximal Ideal - Definition

Definition

There are other equivalent ways of expressing the definition of maximal one-sided and maximal two-sided ideals. Given a ring R and a proper ideal I of R (that is IR), I is a maximal ideal of R if any of the following equivalent conditions hold:

  • There exists no other proper ideal J of R so that IJ.
  • For any ideal J with IJ, either J = I or J = R.
  • The quotient ring R/I is a simple ring.

There is an analogous list for one-sided ideals, for which only the right-hand versions will be given. For a right ideal A of a ring R, the following conditions are equivalent to A being a maximal right ideal of R:

  • There exists no other proper right ideal B of R so that AB.
  • For any right ideal B with AB, either B = A or B = R.
  • The quotient module R/A is a simple right R module.

Maximal right/left/two-sided ideals are the dual notion to that of minimal ideals.

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