minimal ideal
English
Noun
minimal ideal (plural minimal ideals)
- (algebra, ring theory) A nonzero (two-sided) ideal that contains no other nonzero two-sided ideal.
- 1956, Nathan Jacobson, Structure of Rings, American Mathematical Society, page vii:
- Of particular interest are the primitive rings with minimal ideals, algebraic algebras and algebras with a polynomial identity.
- 1970, Mary W. Gray, A Radical Approach to Algebra, Addison-Wesley, page 33:
- Theorem 12. A semisimple ring R has only a finite number of minimal ideals and is their direct sum. Moreover, each minimal ideal is a simple ring.
- 2009, John Rhodes, Benjamin Steinberg, The q-theory of Finite Semigroups, Springer, page 596:
- Note that if a semigroup has a zero, then its minimal ideal is {0}. In this context, the notion of minimal ideal is not so useful, and so we introduce the notion of a 0-minimal ideal.
Coordinate terms
Related terms
- minimal left ideal
- minimal right ideal
Further reading
Ideal on Wikipedia.Wikipedia
Maximal ideal on Wikipedia.Wikipedia
- Minimal ideal on Encyclopedia of Mathematics
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