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I am currently trying to understand different kinds of rings. Is my understanding of the following correct?

Prime elements are always irreducible. The decomposition of a ring element into prime elements is always unique. Irreducible elements are not necessarily prime. If all (non-zero and non-unitary) irreducible elements of a ring are prime elements, we have a unique factorization ring. So in such a ring, also the decomposition into irreducible elements is unqiue.

What's the remarkable thing about this? I don't really see a motivation for distinguishing between irreducible elements and prime elements yet.

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(Caveat: this is all done assuming you mean domain where you wrote ring. This is the natural environment for unique factorization.)

If all (non-zero and non-unitary) irreducible elements of a ring are prime elements, we have a unique factorization ring.

This is incorrect. In addition to irreducibles being prime, you also need the ascending chain condition on principal ideals.

I don't really see a motivation for distinguishing between irreducible elements and prime elements yet.

It is possible for an element to be written as two distinct factorizations of irreducible elements, but this is not true for factorizations using prime elements. The two things are of completely different characters. Is this not reason enough to distinguish?

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