Prime Element
In abstract algebra, 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.
Interest in prime elements comes from the Fundamental theorem of arithmetic, which asserts that each 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.
Prime elements should not be confused with irreducible elements. In an integral domain, every prime is irreducible but the converse is not true in general. However, in unique factorization domains, or more generally in GCD domains, primes and irreducibles are the same.
Being prime is also relative to which ring an element is considered to be in; for example, 2 is a prime element in Z but it is not in Z, the ring of Gaussian integers, since and 2 does not divide any factor on the right.
Read more about Prime Element: Examples
Famous quotes containing the words prime and/or element:
“In time, after a dozen years of centering their lives around the games boys play with one another, the boys’ bodies change and that changes everything else. But the memories are not erased of that safest time in the lives of men, when their prime concern was playing games with guys who just wanted to be their friendly competitors. Life never again gets so simple.”
—Frank Pittman (20th century)
“Every American, to the last man, lays claim to a “sense” of humor and guards it as his most significant spiritual trait, yet rejects humor as a contaminating element wherever found. America is a nation of comics and comedians; nevertheless, humor has no stature and is accepted only after the death of the perpetrator.”
—E.B. (Elwyn Brooks)