minimal ideal

English

Noun

minimal ideal (plural minimal ideals)

1. (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

• maximal ideal

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