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. 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

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Footnotes

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