Hellenica World

# .

In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.

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 | bc for some b and c in R, then a|b or a|c.) In an integral domain, every prime element is irreducible, but the converse is not true in general. The converse is true for unique factorization domains (or, more generally, GCD domains.)

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 the ideal generated by x is a prime ideal of D.
Example

In the quadratic integer ring $$\mathbf{Z}[\sqrt{-5}]$$, 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,

$$3 \mid \left(2 + \sqrt{-5}\right)\left(2 - \sqrt{-5}\right)=9,$$

but 3 does not divide either of the two factors.

Irreducible polynomial

References

Consider p a prime that is reducible: p=ab. Then p | ab \Rightarrow p | a or p | b. Say p | a \Rightarrow a = pc, then we have p=ab=pcb \Rightarrow p(1-cb)=0. Because R is an integral domain we have cb=1. So b is a unit and p is irreducible.
Sharpe (1987) p.54
http://planetmath.org/encyclopedia/IrreducibleIdeal.html

William W. Adams and Larry Joel Goldstein (1976), Introduction to Number Theory, p. 250, Prentice-Hall, Inc., ISBN 0-13-491282-9

Sharpe, David (1987). Rings and factorization. Cambridge University Press. ISBN 0-521-33718-6. Zbl 0674.13008.

Any non-trivial idempotent a is a zero divisor (because ab = 0 with neither a nor b being zero, where b = 1 − a). This shows that integral domains and division rings don't have such idempotents. Local rings also don't have such idempotents, but for a different reason. The only idempotent contained in the Jacobson radical of a ring is 0.

For associative algebras or Jordan algebras over a field, the Peirce decomposition is a decomposition of an algebra as a sum of eigenspaces of commuting idempotent elements.

Another method to obtain a field from a commutative ring R is taking the quotient R / m, where m is any maximal ideal of R. The above construction of F = E[X] / (p(X)), is an example, because the irreducibility of the polynomial p(X) is equivalent to the maximality of the ideal generated by this polynomial. Another example are the finite fields Fp = Z / pZ.