| 词条 | Prime element |
| 释义 |
In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish prime elements from irreducible elements, a concept which is the same in UFDs but not the same in general. DefinitionAn element {{mvar|p}} of a commutative ring {{mvar|R}} is said to be prime if it is not the zero element or a unit and whenever {{mvar|p}} divides {{mvar|ab}} for some {{mvar|a}} and {{mvar|b}} in {{mvar|R}}, then {{mvar|p}} divides {{mvar|a}} or {{mvar|p}} divides {{mvar|b}}. Equivalently, an element {{mvar|p}} is prime if, and only if, the principal ideal {{math|(p)}} generated by {{mvar|p}} is a nonzero prime ideal.[1] (Note that in an integral domain, the ideal {{math|(0)}} is a prime ideal, but {{math|0}} is an exception in the definition of 'prime element'.) Interest in prime elements comes from the Fundamental theorem of arithmetic, which asserts that each nonzero integer can be written in essentially only one way as 1 or −1 multiplied by a product of positive prime numbers. This led to the study of unique factorization domains, which generalize what was just illustrated in the integers. Being prime is relative to which ring an element is considered to be in; for example, 2 is a prime element in {{math|Z}} but it is not in {{math|Z[i]}}, the ring of Gaussian integers, since {{math|1=2 = (1 + i)(1 − i)}} and 2 does not divide any factor on the right. Connection with prime ideals{{Main|Prime ideal}}An ideal {{math|I}} in the ring {{math|R}} (with unity) is prime if the factor ring {{math|R/I}} is an integral domain. A nonzero principal ideal is prime if and only if it is generated by a prime element. Irreducible elements{{Main|Irreducible element}}Prime elements should not be confused with irreducible elements. In an integral domain, every prime is irreducible[2] but the converse is not true in general. However, in unique factorization domains,[3] or more generally in GCD domains, primes and irreducibles are the same. ExamplesThe following are examples of prime elements in rings:
References
1. ^{{harvnb|Hungerford|1980|loc=Theorem III.3.4(i)}}, as indicated in the remark below the theorem and the proof, the result holds in full generality. 2. ^{{harvnb|Hungerford|1980|loc=Theorem III.3.4(iii)}} 3. ^{{harvnb|Hungerford|1980|loc=Remark after Definition III.3.5}}
| authorlink=Thomas W. Hungerford | last=Hungerford | first=Thomas W. | title=Algebra | edition=Reprint of 1974 | year=1980 | publisher=Springer-Verlag | location=New York | series=Graduate Texts in Mathematics | volume=73 | isbn=978-0-387-90518-1 | mr=0600654 }}
1 : Ring theory |
| 随便看 |
|
开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。