maximal ideal

maximal ideal

English

Noun

maximal ideal (plural maximal ideals)

1. (algebra, ring theory) An ideal 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).
   • 1994, William M. McGovern, Completely Prime Maximal Ideals and Quantization, American Mathematical Society, page 15,

     Denote the minimal prime ideal in (b) by ${\\displaystyle J_{\\mathsf {max}}(\\lambda )}$, the unique maximal ideal of infinitesimal character ${\\displaystyle \\lambda }$.

   • 2004, Ayman Badawi, Abstract Algebra Manual: Problems and Solutions, Nova Science, 2nd Edition, page 87,

     Let S be the set of all prime ideals of B, and H be the set of all maximal ideals of B.

   • 2013, Igor R. Shafarevich, Basic Algebraic Geometry 2: Schemes and Complex Manifolds, Springer, 3rd Edition, page 11,

     In particular, a prime ideal ${\\displaystyle {\\mathfrak {p}}\\subset A}$ is a closed point of ${\\displaystyle \\mathrm {Spec} \\ A}$ if and only if ${\\displaystyle {\\mathfrak {p}}}$ is a maximal ideal.

Coordinate terms

• minimal ideal

Related terms

• maximal left ideal
• maximal right ideal

Further reading

• Ideal on Wikipedia.Wikipedia
• Minimal ideal on Wikipedia.Wikipedia
• Maximal ideal on Encyclopedia of Mathematics