I'm playing around and trying to construct rings with different numbers of irreducible elements, hence the above question.
2026-04-02 05:25:17.1775107517
Are there rings with uncountably many irreducible elements (prime elements, if in a PID)?
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Take a polynomial ring in a single variable $x$ with coefficients in real numbers (or complex numbers, what is important is the uncountablity). Then all degree one polynomials $(x-\alpha)$ are irreducible. Being degree 1 they cannot be products of lower degree polynomials! This being a PID irreducible, prime are one and the same.