Example of $\alpha \in R$ s.t. $\alpha$ is not prime and $\alpha$ cannot be factored into a product of primes

Question

Let $R$ be a ring and $\alpha \in R$. Is there any example s.t. (1) $\alpha$ is not a prime (i.e., $\alpha$ is not invertible and $\alpha \neq ab$ for all not invertible elements $a,b \in R$) and (2) $\alpha$ cannot be factored into a product of primes (i.e., there exists $u, p_1,p_2, \dots p_r$ s.t. $\alpha=up_1p_2 \dots p_r$, where $u$ is invertible, $p_i$ is prime)?
My reasoning: $\alpha$ (if exist) must be an infinite products, e.g. $\alpha= u \prod _{i=1} ^{\infty} p_i$, where $u$ is invertible and $p_i$ is prime. Because otherwise: a) $\alpha$ is a product of finite primes, which violates (2). b) $\alpha \neq ab$ for all $a,b \in R$, so $\alpha$ is prime, which violates (1).
Is my reasoning correct? Is there any example? Thank you in advance.
Edit 1: b) is incorrect. The rest choice should be: b') $\alpha \ne ab$ for all prime $a,b \in R$. And it is still possible for $\alpha=ab$ for some non-invertible $a,b \in R$.

Answer 1

Consider the ring of algebraic integers. Every element has a square root, so there are no irreducibles, hence no primes.