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Prime element
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.
==Definition==
An element of a commutative ring is said to be prime if it is not zero or a unit and whenever divides for some and in , then divides or divides . Equivalently, an element is prime if, and only if, the principal ideal generated by is a nonzero prime ideal.〔, as indicated in the remark below the theorem and the proof, the result holds in full generality.〕
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 but it is not in , the ring of Gaussian integers, since and 2 does not divide any factor on the right.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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