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.

Read more about this topic:  Maximal Ideal

Famous quotes containing the word definition:

    Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.
    Nadine Gordimer (b. 1923)

    Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.
    Elaine Heffner (20th century)

    Scientific method is the way to truth, but it affords, even in
    principle, no unique definition of truth. Any so-called pragmatic
    definition of truth is doomed to failure equally.
    Willard Van Orman Quine (b. 1908)