**Prime Elements in Rings**

Prime numbers give rise to two more general concepts that apply to elements of any commutative ring *R*, an algebraic structure where addition, subtraction and multiplication are defined: *prime elements* and *irreducible elements*. An element *p* of *R* is called prime element if it is neither zero nor a unit (i.e., does not have a multiplicative inverse) and satisfies the following requirement: given *x* and *y* in *R* such that *p* divides the product *xy*, then *p* divides *x* or *y*. An element is irreducible if it cannot be written as a product of two ring elements that are not units. In the ring **Z** of integers, the set of prime elements equals the set of irreducible elements, which is

In any ring *R*, any prime element is irreducible. The converse does not hold in general, but does hold for unique factorization domains.

The fundamental theorem of arithmetic continues to hold in unique factorization domains. An example of such a domain is the Gaussian integers **Z**, that is, the set of complex numbers of the form *a* + *bi* where *i* denotes the imaginary unit and *a* and *b* are arbitrary integers. Its prime elements are known as Gaussian primes. Not every prime (in **Z**) is a Gaussian prime: in the bigger ring **Z**, 2 factors into the product of the two Gaussian primes (1 + *i*) and (1 − *i*). Rational primes (i.e. prime elements in **Z**) of the form 4*k* + 3 are Gaussian primes, whereas rational primes of the form 4*k* + 1 are not.

Read more about this topic: Prime Number, Generalizations

### Famous quotes containing the words rings, prime and/or elements:

“‘She has got *rings* on every finger,

Round one of them she have got three.

She have gold enough around her middle

To buy Northumberland that belongs to thee.”

—Unknown. Young Beichan (l. 61–64)

“Few white citizens are acquainted with blacks other than those projected by the media and the so—called educational system, which is nothing more than a system of rewards and punishments based upon one’s ability to pledge loyalty oaths to Anglo culture. The media and the “educational system” are the *prime* sources of racism in the United States.”

—Ishmael Reed (b. 1938)

“In verse one can take any damn constant one likes, one can alliterate, or assone, or rhyme, or quant, or smack, only one MUST leave the other *elements* irregular.”

—Ezra Pound (1885–1972)