Ireducibilné

Ireducibilné is a term used in abstract algebra, particularly in ring theory, to describe certain types of elements or ideals within a ring. An element in a commutative ring with unity is considered ireducibilné if it is not zero, not a unit, and whenever it can be written as a product of two elements, at least one of those elements must be a unit. Essentially, an ireducibilné element cannot be factored into "smaller" non-unit elements within the ring.

The concept is closely related to that of prime elements. In a unique factorization domain (UFD), every non-zero, non-unit ireducibilné element is also prime. However, the converse is not always true; an ireducibilné element is not necessarily prime in all rings. For example, in the ring of integers Z, ireducibilné and prime elements are identical.

In the context of ideals, an ireducibilné ideal I of a commutative ring R is a proper ideal such that if I is contained in a product of two ideals A and B, then I must contain either A or B. This means the ideal cannot be "broken down" into a product of smaller ideals in a significant way. The study of ireducibilné elements and ideals is fundamental to understanding the structure of rings and their factorization properties.