Divisibility (ring theory)

In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers. With the development of abstract rings, of which the integers are the archetype, the original notion of divisor found a natural extension. Divisibility is a useful concept for the analysis of the structure of commutative rings because of its relationship with the ideal structure of such rings. In the ring of integers consider elements . is a divisor of , because with remainder or, phrased differently and more appropriate for generalization, there is such that . is not a divisor of , because with remainder resp. there is no such that . In the field of reals , an extension of the integers, we have is still a divisor of , because resp. there is such that . becomes a divisor of , because is now a real number in resp. there is such that . In the ring of integer polynomials we have that is a divisor of , because by polynomial division with remainder resp. there is such that . is not a divisor of , because with remainder resp. there is no such that . In fact one can show that has no nontrivial divisors, which in this case means in particular that the quadratic polynomial has no linear divisors like . Having no nontrivial divisors is called being irreducible. Expressed symbolically, there are no of which not at least one is a unit of the ring such that . is not a divisor of , because with remainder resp. there is no such that . As before, one can show that has no nontrivial divisors. In the ring of real polynomials , an extension of the integer polynomials, we have is still a divisor of , because resp. there is such that . acquires nontrivial divisors, explicitly , because resp. there is such that . Note that is still not a divisor of , because of the same reason as for above. still has no nontrivial divisors in , i.e. remains irreducible in and acquires no nontrivial divisibility. These examples illustrate that the notion of divisibility does not only depend on the elements and themselves, but also on the context of the algebraic structure of a ring in which and the multiplication operation are considered.