minimal ideal

plural minimal ideals

1. ring theoryalgebraA nonzero (two-sided) ideal that contains no other nonzero two-sided ideal.

   Of particular interest are the primitive rings with minimal ideals, algebraic algebras and algebras with a polynomial identity.

   1956, Nathan Jacobson, Structure of Rings, American Mathematical Society, page vii

   2 more examples

   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.

   1970, Mary W. Gray, A Radical Approach to Algebra, Addison-Wesley, page 33

   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.

   2009, John Rhodes, Benjamin Steinberg, The q-theory of Finite Semigroups, Springer, page 596