Ring theory MOC

Irreducible element

Let be a ring. An element is irreducible iff is not a unit but whenever with then or is a unit. #m/def/ring

This is one way to generalize the Prime number to an arbitrary ring.1

Properties

  1. For an Integral domain, is irreducible iff is maximal among principal ideals.
Proof of 1

Assume is irreducible, and suppose . Then for some , whence either is a unit, implying , or is a unit, implying and thus . Therefore is maximal among principal ideals.

Now suppose is maximal among principal ideals, and let for . Then and . If we are done, so assume . Then and are associate elements and thus for . Hence and thus . Therefore is irreducible, proving ^P1

#state/tidy | #lang/en | #SemBr

Footnotes

  1. 2022. Algebraic number theory course notes, §1.1, p. 1 ↩