Locally maximal ideals are prime in a distributive lattice, but not in a general lattice. The aim of this paper is to prove several necessary and sufficient conditions for a locally maximal ideal in a lattice to be prime. Dual of the concept of relative annihilator in a lattice is used to...
showed that in commutative semigroups (resp. ordered semigroups) with identity, every maximal ideal is a prime ideal and the converse is not true in general. In this note we prove that for an ordered semigroup S satisfying S = ( S 2 ] all maximal ideals are weakly prime ideals. This...
A maximal ideal (x,y) is a type of primary ideal, as it is a prime ideal that is not contained in any other ideal. Additionally, primary ideals can be decomposed into a product of maximal ideals, which is known as the primary decomposition theorem. How are maximal ideals and primary id...
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These ideals are proper if and only if n ≠ 1, and maximal if and only if n > 0 is a prime number. The ideal (0) is prime but is not maximal. By Krull’s theorem (Theorem 2.27), Spec (R) ≠ ∅ if R ≠ {0}. The proof of the following result is left as an exercise*...
1.This paper generalizes the theories of the minimal prime ideals of a ideal, the maximal prime ideals of a ideal in the commutative rings to DQC rings.将交换环中关于一个理想的极小素理想和极大素理想的两种理论推广到DQC一环中,得到了与交换环中相近的重要结果。 延伸阅读 极大理想极大理想maximal ...
Spaces X in which all prime z-ideals of C(X) are minimal or maximal1 Quasi P -spaces are dened to be those Tychono spaces X such that each prime z-ideal of C(X) is either minimal or maximal. This article is devoted to a syst... M Henriksen,RG Woods - 《Commentationes Mathemat...
Banerjee B., Ghosh S.K., Henriksen M.: Unions of minimal prime ideals of rings of continuous functions on compact spaces. Algebra Universalis 62, 239–246 (2009)MathSciNetCrossRef 9. Blair R.L.: Spaces in which special sets are z-embedded. Canad. J.Math. 28, 673–690 (1984)MathSci...
Prime ideals are primary andradical. Maximal ideals arezero-dimensionaland prime. By convention, prime, primary, and maximal ideals must also be proper, meaning that they are not the entire polynomial ring. TheIsPropercommand can be used to test this condition separately. An optional second argum...
1) maximal prime ideal 极大素理想 1. This paper generalizes the theories of the minimal prime ideals of a ideal, the maximal prime ideals of a ideal in the commutative rings to DQC rings. 将交换环中关于一个理想的极小素理想和极大素理想的两种理论推广到DQC一环中,得到了与交换环中相近的重...