Operations in modular arithmetic - addition, multiplication and combinations?

We have the following definition

Let $n\in\mathbb{N}$. We define two binary operations on the set

$$\mathbb{Z}_n=\{0,1,2,\dots,n-1\}$$

1. We call $\oplus:\mathbb{Z}_n\times\mathbb{Z}_n\to\mathbb{Z}_n$, such that $a\oplus b=(a+b)\pmod{n}$ addition modulo $n$
2. We call $\otimes:\mathbb{Z}_n\times\mathbb{Z}_n\to\mathbb{Z}_n$, such that $a\otimes b=(a\cdot b)\pmod{n}$ multiplication modulo $n$

My question is, can we have at the same time an addition and a multiplication modulo $n$, for example something like $g(e)\oplus b=(g(e)+b)\pmod{n}=(a\cdot e+ b)\pmod{n}$? Where $g(e)=a\cdot e$

Also, in case $x=(y+z)\pmod{n}$, then $x-y-z$ is a multiple of $n$. Could we also write $(x-y-z)\pmod{n}=x\ominus_{n}(y\oplus_{n} z)?$

The triplet $(\Bbb Z_n, \oplus, \otimes)$ forms something called "the ring of integers modulo $n$," which is a commutative ring with unity.

Could we also write $(x-y-z)\pmod{n}=x\ominus_{n}(y\oplus_{n} z)?$

Yes.