Show 'a' is Irreducible: If R a Max Ideal in R

[mansi]

Here’s an interesting question…
Let R be a commutative ring and ‘a’ an element in R. If the principal ideal Ra is a maximal ideal of R then show that ‘a’ is an irreducible element.
If a is prime, this is pretty obvious…if a is not prime, then we say a= bc for some b,c in R. Now we need to show that either of them is a unit. I can’t imagine how…

 

[Hurkyl]

If a is reducible, try finding a bigger ideal than (a). Maximal ideals are always prime ideals!

 

[mathwonk]

the usual definition of a prime element in a commutative ring with identity, is that it generates a prime ideal.