Minimal Prime (other)
In mathematics, the term minimal prime may refer to *Minimal prime ideal, in commutative algebra *Minimal prime (recreational mathematics) In recreational number theory, a minimal prime is a prime number for which there is no shorter subsequence of its digits in a given base that form a prime. In base 10 there are exactly 26 minimal primes: : 2, 3, 5, 7, 11, 19, 41, 61, 89, 4 ...

Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ...
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Minimal Prime Ideal
In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal primes. Definition A prime ideal ''P'' is said to be a minimal prime ideal over an ideal ''I'' if it is minimal among all prime ideals containing ''I''. (Note: if ''I'' is a prime ideal, then ''I'' is the only minimal prime over it.) A prime ideal is said to be a minimal prime ideal if it is a minimal prime ideal over the zero ideal. A minimal prime ideal over an ideal ''I'' in a Noetherian ring ''R'' is precisely a minimal associated prime (also called isolated prime) of R/I; this follows for instance from the primary decomposition of ''I''. Examples * In a commutative artinian ring, every maximal ideal is a minimal prime ideal. * In an integral domain, the only minimal prime ideal is the zero ideal. * In the ring Z of integers, the minimal prime ideals ...
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