The difference between Maximal ideal and Minimal ideal
When used as nouns, maximal ideal means an which cannot be made any larger (by adjoining any element to it) without making it improper (i.e., equal to the whole of the containing algebraic structure), whereas minimal ideal means a nonzero (two-sided) ideal that contains no other nonzero two-sided ideal.
check bellow for the other definitions of Maximal ideal and Minimal ideal
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Maximal ideal as a noun (algebra, ring theory):
An which cannot be made any larger (by adjoining any element to it) without making it improper (i.e., equal to the whole of the containing algebraic structure).
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Minimal ideal as a noun (algebra, ring theory):
A nonzero (two-sided) ideal that contains no other nonzero two-sided ideal.