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Ever since learning basic ring theory, I've always felt kind of confused about the fact that:

  • maximal ideals are prime (because every field is an integral domain), but
  • irreducible elements needn't be prime.

Recently, the reason for this suddenly snapped into focus: the problem, in short, is that irreducibility can only "see" principal ideals, and therefore isn't usually strong enough to imply full-blown primeness, except in a principal ideal ring. Indeed:

Proposition. Let $R$ denote a commutative ring. Then for all $x \in R,$ the following are equivalent.

  • $x$ is irreducible.
  • $Rx$ is maximal among all proper principal ideals.

Corollary. Let $R$ denote a principal ideal commutative ring. Then for all $x \in R,$ the following are equivalent.

  • $x$ is irreducible.
  • $Rx$ is maximal among all proper ideals.

Motivated by this realization, I was just wondering:

Question. Is there a name for those elements $x$ of a commutative ring $R$ such that $Rx$ is maximal among all proper ideals?

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These elements are called m-irreducible or i-atomic sometimes. In rings with zero-divisors, these are actually distinct from irreducible elements. I recommend the paper by D.D. Anderson and Valdez Leon called Factorization in Commutative Rings with Zero-divisors if you want to read more. In a domain which is not a field, $(0)$ is not maximal among principal ideals (obviously) despite being irreducible, so the version you state is not quite even true in domains, although it is for non-zero elements.

The paper I suggest gives you some non-trivial examples of all the various relations between the notions of irreducible as well as examples which show the reverse inclusions are not true.

Edit: I figured I would add the example just in case people cannot access that paper. In $\mathbb{Z} \times \mathbb{Z}$, the element $(0,1)$ is prime and therefore irreducible, but not maximal among principal ideals since it is properly contained in $(2,1)$.

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