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Corollary 1. If $R$ is an atomic domain with at least two distinct irreducible elements, then $N_R = R^{\times}.$

Proof. As $R$ has at least two distinct irreducible elements, $N_R$ cannot contain any irreducible element. Since $N_R$ is saturated and $R$ is atomic, $N_R$ cannot contain any non-unit element.

Corollary 1. If $R$ is an atomic domain with at least two non-associated irreducible elements, then $N_R = R^{\times}.$

Proof. As $R$ has at least two non-associated irreducible elements, $N_R$ cannot contain any irreducible element. Since $N_R$ is saturated and $R$ is atomic, $N_R$ cannot contain any non-unit element.

Proof. Cleary, $R[X]$ is not a field. Let $f(X, Y) \in (R[X])[Y]$ be a non-constant polynomial with coefficients in $R[X]$. Then $f(X, X^n)$ is a non-constant polynomial with coefficients in $R$ for all $n$ large enough. Therefore $f(X, Y)$ isn't a $u$-polynomial over $R[X]$. Consequently, $u$-polynomials over $R[X]$ are constant polynomials and the result follows from Claim 4.

Corollary 2. Assume that $R$ is domain satisfying the ascending chain condition on principal ideals (ACCP) and such that $S_R = N_R$, e.g, $R$ is a Noetherian local domain. Then either $R$ is a discrete valuation ring (DVR) and $S_R = R \setminus \{0\}$ or $S_R = R^{\times}$.

Proof. Assume that $S_R$ contains a non-unit element. As $R$ is atomic and $S_R$ is saturated, the set $S_R$ contains an irreducible element $p$. Since $p \in N_R$, the element $p$ divides any non-unit of $R$. Therefore $Rp$ is the unique maximal ideal of $R$. Take $a \in I \Doteq \bigcap_n Rp^n$ and write $a = a_n p^n$ for every $n > 0$. If a is not zero, then $(Ra_n)_n$ defines a strictly increasing sequence of principal ideals, contradicting (ACCP). Thus $I = 0$ and it easily follows that $R$ is a DVR.

Erratum. I claimed in an earlier version of this answer that the conclusion of Corollary 2 holds if the condition (ACCP) is replaced by "$R$ is atomic". As I ignore whether this is true or false (the condition (ACCP) is known to be strictly stronger than atomicity for a domain), I changed Corollary 2 accordingly. I thank @users for pointing out this issue.

Corollary 2. Assume that $R$ is an atomic domain such that $S_R = N_R$, e.g, $R$ is a Noetherian local domain. Then either $R$ is a discrete valuation ring (DVR) and $S_R = R \setminus \{0\}$ or $S_R = R^{\times}$.

Proof. Assume that $S_R$ contains a non-unit element. As $R$ is atomic and $S_R$ is saturated, the set $S_R$ contains an irreducible element $p$. Since $p \in N_R$, the element $p$ divides any non-unit of $R$. Therefore $Rp$ is the unique maximal ideal of $R$. Since an irreducible element of $R$ lies necessarily in $Rp$, $p$ is the unique irreducible element of $R$, up to multiplication by a unit. Since $R$ is atomic $R$, we deduce that $R$ is a DVR.