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
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