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(1) Is there a finite nilpotent ring $R$ such that the ratio $$\frac{|\{(x,y)\in R\times R \; | \; xy=yx\}}{|R|}$$ is not integer?

Edit: The nilpotent condition is put later.

(2) Is there a finite nilpotent Lie algebra $L$ such that the ratio $$c(L)=\frac{|\{(x,y)\in L\times L \; | \; [x,y]=0\}}{|L|}$$ is not integer?

Edit: The nilpotent condition on the Lie Algebra is put later.

The motivation is that the same question for finite groups has negative answer.

(1) Is there a finite nilpotent ring $R$ such that the ratio $$c(R)=\frac{|\{(x,y)\in R\times R \; | \; xy=yx\}}{|R|}$$ is not integer?

Edit 1: The nilpotent condition is put later.

Edit/Answer: Answer is negative as Frieder Ladisch nicely observed below. That is $c(R)$ is always an integer. Actually $c(R)$ is the number of conjugacy classes of the group $1+R$.

(2) Is there a finite nilpotent Lie algebra $L$ such that the ratio $$c(L)=\frac{|\{(x,y)\in L\times L \; | \; [x,y]=0\}}{|L|}$$ is not integer?

Edit: The nilpotent condition on the Lie Algebra is put later.

The motivation is that the same question for finite groups has negative answer.

(1) Is there a finite nilpotent ring $R$ such that the ratio $$\frac{|\{(x,y)\in R\times R \; | \; xy=yx\}}{|R|}$$ is not integer?

Edit: The nilpotent condition is put later.

(2) Is there a finite nilpotent Lie algebra $L$ such that the ratio $$\frac{|\{(x,y)\in L\times L \; | \; [x,y]=0\}}{|L|}$$ is not integer?

Edit: The nilpotent condition on the Lie Algebra is put later.

The motivation is that the same question for finite groups has negative answer.

(1) Is there a finite nilpotent ring $R$ such that the ratio $$\frac{|\{(x,y)\in R\times R \; | \; xy=yx\}}{|R|}$$ is not integer?

Edit: The nilpotent condition is put later.

(2) Is there a finite nilpotent Lie algebra $L$ such that the ratio $$c(L)=\frac{|\{(x,y)\in L\times L \; | \; [x,y]=0\}}{|L|}$$ is not integer?

Edit: The nilpotent condition on the Lie Algebra is put later.

The motivation is that the same question for finite groups has negative answer.

(1) Is there a finite ring $R$ such that the ratio $$\frac{|\{(x,y)\in R\times R \; | \; xy=yx\}}{|R|}$$ is not integer?

(2) Is there a finite Lie algebra $L$ such that the ratio $$\frac{|\{(x,y)\in L\times L \; | \; [x,y]=0\}}{|L|}$$ is not integer?

The motivation is that the same question for finite groups has negative answer.

(1) Is there a finite nilpotent ring $R$ such that the ratio $$\frac{|\{(x,y)\in R\times R \; | \; xy=yx\}}{|R|}$$ is not integer?

Edit: The nilpotent condition is put later.

(2) Is there a finite nilpotent Lie algebra $L$ such that the ratio $$\frac{|\{(x,y)\in L\times L \; | \; [x,y]=0\}}{|L|}$$ is not integer?

Edit: The nilpotent condition on the Lie Algebra is put later.

The motivation is that the same question for finite groups has negative answer.