How do you distinguish rings with unity?

Recently, I have changed my vocabulary in abstract algebra.

If one doesn't require "ring" to have a unity, then one can say that "ring with unity $1\neq 0$"

However, I changed my vocabulary to call "a ring without unity" a rng, and "ring with unity" a ring.

With this vocabulary, how do I call "rings with unity $1\neq 0$"? Non-trivial ring I guess?

I want to know whether there is a common terminology to call that:)

The zero ring, $\{0\}$, is the unique ring in which the additive identity $0$ and multiplicative identity $1$ coincide. $\{0\}$ is also called the trivial ring. All other rings with $1\ne 0$, are usually just called "rings", but sure they're nontrivial by supposing $1\ne 0$. I guess, depending on context, it might be advantageous to say a priori, whether the ring you're talking about need or need not be nontrivial.