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 I ≠ R), 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 I ⊊ J.
- For any ideal J with I ⊆ J, 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 A ⊊ B.
- For any right ideal B with A ⊆ B, 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:
“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)
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)