algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...

, an irreducible element of a

domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Do ...

is a non-zero element that is not

invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...

(that is, is not a

unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...

), and is not the product of two non-invertible elements.

Relationship with prime elements

Irreducible elements should not be confused with prime elements. (A non-zero non-unit element

a

in a commutative ring

R

is called prime if, whenever

a \mid bc

for some

b

and

c

R,

then

a \mid b

a \mid c.

) In an

integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...

, every prime element is irreducible,Sharpe (1987) p.54 but the converse is not true in general. The converse is true for unique factorization domains (or, more generally,

GCD domain In mathematics, a GCD domain is an integral domain ''R'' with the property that any two elements have a greatest common divisor (GCD); i.e., there is a unique minimal principal ideal containing the ideal generated by two given elements. Equivalentl ...

s). Moreover, while an ideal generated by a prime element is a prime ideal, it is not true in general that an ideal generated by an irreducible element is an irreducible ideal. However, if

D

is a GCD domain and

x

is an irreducible element of

D

, then as noted above

x

is prime, and so the ideal generated by

x

is a prime (hence irreducible) ideal of

D

Example

In the quadratic integer ring

\mathbf sqrt

it can be shown using norm arguments that the number 3 is irreducible. However, it is not a prime element in this ring since, for example, :

3 \mid \left(2 + \sqrt\right)\left(2 - \sqrt\right)=9,

but 3 does not divide either of the two factors.William W. Adams and Larry Joel Goldstein (1976), ''Introduction to Number Theory'', p. 250, Prentice-Hall, Inc.,

References

* {{DEFAULTSORT:Irreducible Element Ring theory Algebraic properties of elements

Relationship with prime elements

Example

See also

References